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#### Differential geometry
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#### Mapping space
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# Contents
* automatic table of contents
{: toc}

# Loops with Singularities # {: #title}

##### Andrew Stacey ##### {: #author}

+-- {: .num_section #sectiona }
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## The Original Question ## {: .num_section}

On [MathOverflow](http://www.mathoverflow.com) there was a question "[I was wondering if the set of singular loops is somewhere a submanifold of loop space?](http://mathoverflow.net/questions/75071/i-was-wondering-if-the-set-of-singular-loops-is-a-somewhere-submanifold-of-loop)". [[Ryan Budney]] and [[Torsten Ekedahl]] gave two answers that showed that the answer was "No". In the chronology, Torsten's answer came first and gave an example to show why the space described in the first version of the question could not work. The question was modified to take account of that, whereupon Ryan stepped in to show why the modified space could not be a smooth manifold. Whilst Ryan and Torsten's answers do give the complete answer, the question intrigued [[Andrew Stacey|me]] since there are spaces related to the one in the question that are smooth manifolds. This page is the result of thinking about that.

At time of writing, the question reads as follows (formatting and links added).

> The set of all smooth maps $S^{1}\to M^{n}$ ($M$ is a smooth manifold) is a generalized manifold (see [[smooth loop space]]).

> I was wondering if the set of singular loops (maps with self-crossings or zeros of derivative) is a ([[Fréchet manifold|Fréchet]], [[Frölicher space|Frölicher]], [[diffeological space|diffeological]]) submanifold of the [[loop space]]?

> **EDIT:** it is clear that answer is no (see [Torstens answer](http://mathoverflow.net/questions/75071/i-was-wondering-if-the-set-of-singular-loops-is-a-somewhere-submanifold-of-loop/75077#75077)). So, I rewrite the question: is it true that set of singular loops is a collection of submanifolds of different codimension (set of loops with one self-intersection has codimension $\dim M-2$, loops with zero of derivative has codimension $\dim M -3$ and so on. Set of loops with infinite number of singularities should have infinite codimension\dots . Let's forget about them.)

> So, the question is about local situation: for example, let's consider a loop with one self-intersection (and without other singlarities). Is it true that set of near loops with one self-intersection is a submanifold in sense of ([[Fréchet manifold|Fréchet]], [[Frölicher space|Frölicher]], [[diffeological space|diffeological]])?

> **EDIT 2.** I reformulate the question. $\map (S^{1} \to \mathbb{R} ^{3})$ is a functional space, so we can apply a technique of singularity theory. Generic map $f$ with one self-intersection has one-parameter versal deformation $V\colon [0,1]\times S^{1}\to \mathbb{R} ^{3}$ -- and any deformations are induced from $V$. Does it imply that $D$ (set of singular loops) near the $f$ is a submanifold of codimension 1 in sense of [[Fréchet manifold|Fréchet]] or [[Frölicher space|Frölicher]]?

I'm going to take a different approach to this question. I'm going to start with a space that I know is a submanifold of $L M$ and modify it little-by-little until I get to what (I think) you describe. As I do so, hopefully it will be clear where we lose the manifold structure (and why).

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## Finding a Manifold ## {: .num_section}

The space that is a submanifold is the space of loops with a particular coincidence. Let us think of the circle as the unit circle in the complex plane (for definiteness). Then of all the smooth maps $S^{1} \to M$ we can consider the subset which are coincident at $1$ and $-1$. Identifying $1$ and $-1$ gives us one form of the figure eight and so we can think of this subspace as $C^{\infty } (8,M)$. Indeed, if we work in one of the categories of [generalised smooth space](http://ncatlab.org/nlab/show/generalised smooth space), then there is a "smooth space" $8$ and the subspace we are interested in truly is $C^{\infty } (8,M)$.

This space is a manifold, in whatever sense you wish. The model space is $C^{\infty } (8,\mathbb{R} ^{n})$; its tangent space at a map $f$ is the space of sections of $f^{*} T M$ which agree at $1$ and $-1$. It is also a submanifold of $L M$, and actually has a tubular neighbourhood. So it's as nice as you can get!

The most obvious difference between this and your space is that we have not just a coincidence, but we have fixed where it must occur. So our first generalisation is to allow that coincidence to move. But we want to proceed cautiously, so we start by considering triples $(f,p,q)$ where $f \colon S^{1} \to M$ is a smooth map and $p,q \in S^{1}$ are two points. This is $L M \times S^{1} \times S^{1}$. Next, we consider the subset of these triples that satisfy $f(p) = f(q)$ and $p \ne q$.

I claim that this is a manifold. Let us start by considering those triples $(f,p,q)$ with $\Re p \gt \frac{1}{2}$ and $\Re q \lt - \frac{1}{2}$ (and $f(p) = f(q)$). Let us label this space as $W$. Let us choose a smooth family of diffeomorphisms of the circle, say $\tau _{a,b}$, parametrised by $a, b \in S^{1}$ with $\Re a \gt \frac{1}{2}$ and $\Re b \lt -\frac{1}{2}$ with the property that $\tau _{a,b}(a) = 1$ and $\tau _{a,b}(b) = -1$. We define a map $W \to C^{\infty } (8,M) \times S^{1} \times S^{1}$ by $(f,p,q) \mapsto (f \circ \tau _{p,q}^{-1}, p, q)$. The claim is that this is a diffeomorphism onto its image which is an open subset of the target. The inverse is given by $(f,p,q) \mapsto (f \circ \tau _{p,q},p,q)$. The target is $\{ (f,p,q) : \Re p \gt \frac{1}{2}, \Re q \lt -\frac{1}{2}\} $.

For a more general point, $(f,p,q)$, we choose a diffeomorphism, say $\sigma $, of the circle which maps $p$ to $1$ and $q$ to $-1$. Then we define a diffeomorphism of $L M \times S^{1} \times S^{1}$ by $(g,r,s) \mapsto (g \circ \sigma ^{-1},\sigma (r),\sigma (s))$. This takes $(f,p,q)$ into $W$ and we can pullback the chart from there.

A very simple thing to do at this point is to forget the order of the pair $(p,q)$. This is a $\mathbb{Z} _{2}$--action and the action has all the required properties for the quotient to still be a manifold. Thus we have a manifold of things of the form $(f,\{ p,q\} )$ where $f \colon S^{1} \to M$ is a smooth map, $\{ p,q\} \subseteq S^{1}$ is a two-element subset, and $f(p) = f(q)$ (or $f \!\!\mid _{\{ p,q\} }$ is constant).

+-- {: .num_section #sectionc }
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## Finding a Non-Manifold ## {: .num_section}

The final step is to try to project this down on to the first component. That is, to consider the map from our space to $L M$ given by mapping $(f,\{ p,q\} ) \mapsto f$. What would be nice would be if this were some sort of submersion and we could deduce from it (maybe with a little work) that the image was a manifold.

The first problem that we encounter is that the fibre does not have a constant model. For "most" loops in the image, the preimage consists of a single point. This is because for "most" loops with a coincidence, there is only one such coincidence. But a loop with multiple coincidences has several preimages, one for each coincidence (or, more precisely, one for each two-element subset of each coincidence set).

A loop with multiple coincidences doesn't seem a major problem as we can locally just pick one to label. But that's not actually as simple as it might seem. If we have two coincidences and label one of them, then we can rip apart that coincidence *whilst staying in our space* because the other coincidence keeps us there. But if we destroy the labelled coincidence, what do we do with our label?

This is Torsten's objection, except with multiple double coincidences instead of a single triple one.

To fix this, we try restricting to the subset where there is only one coincidence. This means that our fibres are singletons and we have a smooth bijection from the suitable set of $(f,\{ p,q\} )$ to the set of corresponding loops. Leaving aside the question as to whether or not this is a diffeomorphism, this would seem to give a positive answer to the question.

It doesn't.

The reason is simple, but perhaps a little surprising. That is that the subset of those $(f,\{ p,q\} )$ where $\{ p,q\} $ is the *only* coincidence of $f$ is *not* an open subset of our manifold. Why this might be surprising is because if we go one level down then the corresponding family is open. That is, if we take the subset of those loops with *no* coincidences then this is an open subset of all loops (whence a submanifold).

We can show that this subset is not open by taking a sequence of loops with a double coincidence (one of which is labelled) which converges to a loop with a single coincidence. If it were open, at some point in the sequence all terms would have to have a single coincidence.

This is Ryan's objection.

We cannot fix this. The reason being that this subset *contains* an open subset of our manifold (again, by Ryan's answer) so the only way it could be a submanifold would be if it were open.

So the space in question is most emphatically not a submanifold. But there is a nearby manifold, which is the space of loops with a marked coincidence (and possibly other coincidences). For most loops, these two spaces are the same nearby, but for loops such as those that Torsten and Ryan consider, they aren't.

[[!redirects manifold structure of singular loops]]
[[!redirects manifold structures of singular loops]]
[[!redirects the manifold structure of singular loops]]
[[!redirects on the manifold structure of singular loops]]
[[!redirects manifold structure on the space of singular loops]]
[[!redirects manifold structures on the space of singular loops]]
[[!redirects manifold structures on spaces of singular loops]]
[[!redirects singular loop space]]
[[!redirects singular loop spaces]]


On the Space-Theory of Matter

* [[William Clifford]]:

  \linebreak

  **On the Space-Theory of Matter**

  \linebreak

  Proceedings of the Cambridge Philosophical Society **2** (1864-1876 - Printed 1876) 157-158

  \linebreak

  reprinted in: *The Concepts of Space and Time* Boston Studies in the Philosophy of Science **22** Springer (1976) [[doi:10.1007/978-94-010-1727-5_50](https://doi.org/10.1007/978-94-010-1727-5_50)]

on a speculation quite akin to what later was called *[[geometrodynamics]]* by [[John Wheeler]].

## References

* Wikipedia, _[On the Space-Theory of Matter](http://en.wikisource.org/wiki/On_the_Space-Theory_of_Matter)_

category: reference


On Vortex Atoms

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The text

* [[Lord Kelvin]],

  **On Vortex Atoms**

  Proceedings of the Royal Society of Edinburgh, Vol. VI, pp. 94-105.

  1867

  ([web](http://zapatopi.net/kelvin/papers/on_vortex_atoms.html))

presented the hypothesis, developed with [[Peter Tait]], that [[atoms]] ([[elementary particles]]) are fundamentally knotted [[vortices]] in some [[spacetime]]-filling substance (the "aether"), specifically that the [[chemical elements]] were classified by [[knot]]-types.

#Contents#
* table of contents
{:toc}

## Impact

As a literal theory of [[physics]] the vortex atom hypothesis lasted no more than 30 years, with Thomson himself giving up on it by 1890 ([Kragh 02, p. 34](#Kragh)).
But the vortex atom hypothesis prompted [[Peter Tait]] to begin the classification of [[knots]], initiating what became the active modern [[mathematics|mathematical]] field of [[knot theory]].

This history has at times been invoked as a cautionary tale, to warn against contemporary speculations in [[physics]], such as [[string theory]], which may also seem to be mathematically compelling but with uncertain vindication by [[experiment]].

However, on closer inspection there are some ironic twist and turns to this reading:

### Similarity to the Skyrme model for baryons
 {#SimilarityToSkyrmeModel}

First, recall that in contemporary understanding (but see below), the [[chemical elements]] are (not knots in the aether, as Kelvin hypothesized, but ) types of [[atomic nuclei]], which are [[bound states]] of [[protons]] and [[neutrons]] that, in turn, are thought to be [[bound states]] of "[[constituent quarks]]" (under the "[[strong nuclear force]]"), supposedly described by [[quantum chromodynamics]] (QCD).

The irony here is that -- due to the unresolved "[[confinement]]"- or "[[mass gap problem]]" -- *no* good theoretical derivation of these -- or any other [[hadron|hadronic]] [[bound states]] -- from [[constituent quarks]] in [[QCD]] is available! The [[constituent quark]]-hypothesis works to some extent in [[computer experiment]] (see e.g. [Durr et al. 2009](lattice+gauge+theory#Durr09)), but basic properties even of the most common [[baryons]] such as the [[proton]] remain unexplained (see e.g. the [[proton spin crisis]] and the [[proton radius problem]]); and experiments such as the [[LHC]] keep detecting a long list of novel hadrons which have neither been predicted nor understood by [[QCD]] -- see e.g. [Koppenburg 2021](confinement#Koppenburg21), who accurately reflects the prevalent attitude to this state of affairs by asserting that:

> "[Hadron] properties follow from yet unsolved mysteries of the strong interaction" [sic]

In view of this rather disappointing reality, caution may rather be due with respect to dismissing alternative fundamental approaches out of hand.

Indeed, a fairly successful alternative theory for [[baryon]] [[bound states]] exists -- the *[[Skyrme model]]* -- which arguably is conceptually closer to the intuition behind Kelvin's "vortex atoms" than to the [[quark]]-model -- the main difference being "just" the [[dimension|dimensionality]] and variance of the "knotted structure":

Where a [[knot]] (in Kelvin's proposal) is the [[isotopy class]] of a [[map]] from the [[circle]] (hence the "[[1-sphere]]") into 3d [[Euclidean space]] ([[one-point compactification|1-point compactified]] to the [[3-sphere]]), so a [[Skyrmion]] has underlying it the [[homotopy class]] of a [[map]] from 3d space/[[3-sphere]] to the [[Lie group|group manifold]] of [[SU(2)]], which [[diffeomorphism|is]] itself a 3-sphere.  Concretely, the [[homotopy class]] of this [[Skyrmion]]-map models the global higher-dimensional "knotting" in the [[pion]] [[field (physics)|field]], effectively witnessing each [[baryon]] state as a *[[soliton]]* in a spacetime-filling [[pion]]-field, with its baryon number -- hence its position in the [[periodic table of chemical elements]] -- being the [[winding number]] of the map!

(Notice that it is the [[codomain]] 3-sphere which is fixed by the physics in this picture, not the domain: On a general [[spacetime]]-manifold other than [[Minkowski spacetime]], the [[Skyrmion]]-number will be in its 3-[[Cohomotopy]]-set, [[vanishing at infinity]].)

Therefore, it is not hard to imagine that Kelvin would have embraced this [[Skyrmion]] picture of [[atoms]] as confirming the basic intuition behind his proposal. To the extent that the [[Skyrmion]]-model is indeed a correct description of [[baryons]], the conclusion could be that Kelvin's proposal suffered not from having too much guidance by abstract mathematical argument, but too little (missing the full power of [[homotopy theory]] beyond plain [[knot theory]]). Indeed, according to [Ranada-Trueba 01, p. 200](#RanadaTrueba01):

> [[Tony Skyrme|Skyrme]] had studied with attention Kelvin's ideas on vortex atoms.

To add to the irony, it looks like the [[Skyrme model]] comes to itself (only) in a context of [[Randall-Sundrum model|Randall-Sundum]]-like non-compact [[Kaluza-Klein theory]] and/or [[string theory]], where it finds both refinement compatible with [[experiment]] as well as conceptual explanation, potentially connecting it, logically, to all the rest of [[physics]]:

Namely, the original model proposed by [Skryme 1962](skyrmion#Skyrme62) provided a decent qualitative match to low-lying [[baryons]], but the predicted [[binding energies]] quickly deviated from observation as the baryon number increased. This was eventually fixed (via an observation due to [Atiyah & Manton 1989](skyrmion#AtiyahManton89)) in a rather remarkable way: by postulating that Skyrmions are in fact the "non-compact" [[KK-compactification]] of [[instantons]] in a higher-dimensional [[gauge theory]] ([[D=5 Yang-Mills theory]]):

In this picture, the lowest [[KK-compactification|KK-mode]] of the [[instantons]] of the higher dimensional "[[hidden local symmetry|hidden]]" [[gauge field]] reproduces Skyrme's original model, while the higher KK-modes provide corrections (corresponding to adjoining [[vector meson]]-contributions to the [[pion]]-field) that incrementally move the predicted [[binding energies]] ever closer to the observed values ([Naya & Sutcliffe 2918a](skyrmion#NayaSutcliffe18a), [2018b](skyrmion#NayaSutcliffe18b), for more see the references at *[hadrons as KK-modes of 5d Yang-Mills theory](skyrmion#ReferencesHadronsAsKKModesOf5dYangMillsTheory)*).

Moreover, this higher-dimensional refinement of the [[Skyrme model]] naturally appears within [[intersecting brane models]] within [[string theory]], particularly so in the [[Witten-Sakai-Sugimoto model]] for realizing [[quantum chromodynamics]] on branes in [[string theory]]. In this context the Skyrme model has become part of a broader approach of providing theoretical first-principles foundations for [[non-perturbative field theory|non-perturbative]] [[confinement|confined]] [[quantum chromodynamics]] (including not just [[baryons]] but also [[mesons]], [[glueballs]], etc.), now often known as [[holographic QCD]], or similar.

(In final irony, these results remain almost as underappreciated among [[string theory|string theorists]] as among the [[particle physics|particle]]/[[high energy physics]]-community at large.)

Further reading along these lines may be found in [Rho & Zahed 2016](skyrmion#RhoZahed16) and [Manton 2022](skyrmion#Manton22).

\linebreak

### Similarity with Descartes' thoughts
 {#SimilarityWithDescartesThoughts}

According to the _Routledge encyclopedia of Philosophy_ [here](http://www.rep.routledge.com/article/DA026SECT11):

> [[Descartes]] also rejected atoms and the void, the two central doctrines of the atomists, an ancient school of philosophy whose revival by Gassendi and others constituted a major rival among contemporary mechanists. Because there can be no extension without an extended substance, namely body, there can be no space without body, Descartes argued. His world was a plenum, and all motion must ultimately be circular, since one body must give way to another in order for there to be a place for it to enter ( Principles II: §§2–19, 33). Against atoms, he argued that extension is by its nature indefinitely divisible: no piece of matter in its nature indivisible ( Principles II: §20). Yet he agreed that, since bodies are simply matter, the apparent diversity between them must be explicable in terms of the size, shape and motion of the small parts that make them ( Principles II: §§23, 64) (see [[G. W. Leibniz]] §4 ).

However, according to ([Kragh 02, p. 33](#Kragh)):

>In spite of the similarities, there are marked differences between the Victorian theory and Descartes's conception of matter. Thus, although Descartes's plenum was indefinitely divisible, his ethereal vortices nonetheless consisted of tiny particles in whirling motion. It was non-atomistic, yet particulate. Moreover, the French philosopher assumed three different species of matter, corresponding to emission, transmission, and reflection of light (luminous, "subtle", and material particles). The vortex theory, on the other hand, was strictly a unitary continuum theory.

## Similarity to concepts of modern particle physics
 {#SimilarityToParticlePhysics}

### Baryogenesis

It is however striking that the modern concept of [[baryogenesis]] via the [[chiral anomaly]] and its sensitivity to [[instantons]] is not too far away from Kelvin's intuition.

To play this out in the most pronounced scenario, consider, for the sake of it, a [[no boundary proposal|Hartle-Hawking no-boundary spacetime]] carrying $N$ [[Yang-Mills instantons]]. Notice that an [[instanton]] is in a precise sense the modern higher dimensional and [[gauge theory|gauge theoretic]] version of what Kelvin knew as a fluid [[vortex]].

Then the non-[[conservation law]] of the [[baryon]] [[conserved current]] due to the [[chiral anomaly]] says precisely the following: the net baryon number in the early universe is a steadily increasing number -- this is the modern mechanism of [[baryogenesis]] -- such that as one approaches the asymptotically late time after the "[[Big Bang]]" this number converges onto the integer $N$, the number of instantons.

Hence while in the modern picture of baryogenesis via [[gauge anomaly]] an [[elementary particle]] is not literally identified with an instanton, nevertheless each instanton induces precisely one net [[baryon]].

(If one doesn't want to consider a [[no boundary proposal|Hartle-Hawking-type]] Euclidean no-boundary spacetime but instead a [[globally hyperbolic spacetime]] then the conclusion still holds, just not relative to vanishing baryon number at the "south pole" of the cosmic 4-sphere, but relative to the net baryon  number at any chosen spatial reference slice. )

For more on this see at _[baryogenesis -- Exposition](baryogenesis#Exposition)_.

## Related entries

* [[William Clifford]], _[[On the Space-Theory of Matter]]_

* [[Skyrmion]]

## References

Review and discussion:

* {#Kragh} Helge Kragh, *The Vortex Atom: A Victorian Theory of Everything*, Centaurus **44** 1-2 (2002) 32-114 [[philpapers:KRAVTA](https://philpapers.org/rec/KRATVA), [doi:10.1034/j.1600-0498.2002.440102.x](https://doi.org/10.1034/j.1600-0498.2002.440102.x)]

* {#Corfield} [[David Corfield]], _Smoke Rings_, ([[CorfieldVortexAtoms.pdf:file]])

* [[Samuel Lomonaco]], _The modern legacies of Thomson's atomic vortex theory in classical electrodynamics_, in: [[Louis Kauffman]] (ed.) _The Interface of Knots and Physics_, Proceedings of Symposia in Applied Mathematics, Volume 51 (1996) ([pdf](http://www.csee.umbc.edu/~lomonaco/kelvin/kelvin23.pdf), [doi:10.1090/psapm/051](https://doi.org/10.1090/psapm/051))

Discussion in comparison to [[knot|knotted]] states in [[superconductors]]:

* Filipp N. Rybakov, Julien Garaud, Egor Babaev, _Kelvin knots in superconducting state_, Phys. Rev. B 100, 094515 (2019) ([arXiv:1807.02509](https://arxiv.org/abs/1807.02509))

Discussion in relation to [[skyrmions]]:

* {#RanadaTrueba01} Antonio Ranada, J. Trueba, _Force Lines, Vortex Atoms, Topology, and Physics_, Section I.A In: _Topological Electromagnetism with hidden nonlinearlity_ in: Myron Evans (ed.) _Modern Nonlinear Optics -- Part 3_, Wiley 2001 ([doi:10.1002/0471231495.ch2](https://doi.org/10.1002/0471231495.ch2))

category: reference


on-shell recursion

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## Idea

A method for computing [[scattering amplitudes]] in [[QFT]] and in [[string theory]].

## Related entries

* [[shell]]
* [[string theory results applied elsewhere]]

## References

### In quantum field theory

Reviews include

* [[Matthew Strassler]], _From string theory to the large hadron collider_ ([blog post](http://profmattstrassler.com/2012/08/15/from-string-theory-to-the-large-hadron-collider/))

Original articles include

* [[Zvi Bern]], [[Lance Dixon]], [[David Kosower]], _On-Shell Methods in Perturbative QCD_ ([arXiv:0704.2798](http://arxiv.org/abs/0704.2798))

### In string theory

* Rutger Boels, _On-shell recursion for string theory amplitudes on the disk and the sphere_ ([pdf](http://www.itp.uni-hannover.de/nordic/slides/Boels.pdf))


one

#Contents#
* table of contents
{:toc}

## Idea

The multiplicative [[unit]] in the [[natural numbers]] $\mathbb{N}$, [[integers]] $\mathbb{Z}$, [[real numbers]] $\mathbb{R}$ and [[complex numbers]] $\mathbb{C}$ is pronounced **one** and denoted "$1$".  Originally, in [[Euclid]]'s *[[Elements]]*, Book VII, this element $1 \in \mathbb{N}$ is called a *mονάς* (cf. [here](monad+terminology#Euclid) at *[[monad terminology]]*).

More generally, in any [[group]], or even any [[monoid]], the group operation is often called 'multiplication' and written as juxtaposition, and then the [[neutral element]] $\mathrm{e}$ is sometimes also denoted "$1$".  As a consequence, in any [[ring]], or more generally any [[rig]], the two binary operations are called 'addition' and 'multiplication', and the identity for multiplication is called **one**.

[[vertical categorification|Categorifying]] this idea, in any [[monoidal category]] the [[unit object]] may be called **one**, best denoted now by a bolder font, such as $\mathbb{1}$.  This is especially true in the case of a [[cartesian monoidal category]], that is a category with (at least finitary) [[products]].  In this case the [[terminal object]], which serves as the identity for products, is often denoted "$1$":

$$x \times 1 \cong x$$

For example, in the [[category theory]]-literature the [[singleton set]] is often denoted $1$.

Categorifying [[horizontal categorification|horizontally]] instead, any [[identity morphism]] may be thought of as an analog of one and is often written $1$.

All these ideas can be, and have been, categorified further.

## Related concepts

* [[numbers]]

* [[zero]]

* [[two]]

* [[unit]]

* [[unit object]]

* [[monad terminology]]

[[!redirects one]]
[[!redirects 1]]


one-parameter semigroup

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{:toc}

## Idea

A __one-parameter group__ (of [[unitary operators]] in a [[Hilbert space]]) is a [[homomorphism of groups]]
$$\mathbf{R} \to U(H),$$
where $H$ is a [[Hilbert spaces]], [[U(H)|$U(H)$]] denotes its group of [[unitary operators]] and $\mathbf{R}$ the [[addition|additive]] group of [[real numbers]].

More generally, one can define __one-parameter [[semigroups]]__ of operators in a [[Banach space]] $X$ as homomomorphisms of [[monoids]]
$$\mathbf{R}_{\ge0} \to B(X),$$
where $B(X)$ denotes the semigroup of [[bounded operators]] $X\to X$.

Typically, we also require a [[continuous map|continuity]] condition such as continuity in the [[strong topology]].

## Stone theorem

Strongly continuous one-parameter unitary groups $(U_t)_{t\ge0}$ of operators in a Hilbert space $H$
are in [[bijection]] with [[self-adjoint operator|self-adjoint]] [[unbounded operators]] $A$ on $H$:

This bijection sends
$$
  A
    \mapsto
  \big(
     t\mapsto \exp(itA)
  \big)
  \,.
$$

The operator $A$ is [[bounded operator|bounded]] if and only if $U$ is norm-continuous.

## Hille–Yosida theorem

Strongly continuous one-parameter semigroups $T$ of bounded operators on a [[Banach space]] $X$ (alias __$C_0$-semigroups__) satisfying $\|T(t)\|\le M\exp(\omega t)$ are in bijection with closed operators $A\colon X\to X$ with dense domain such that any $\lambda\gt \omega$ belongs to the resolvent set of $A$
and for any $\lambda\gt\omega$ we have
$$\|(\lambda I-A)^{-n}\|\le M (\lambda-\omega)^{-n}.$$

## Related concepts

* [[exponential]]

## References

[…]

See also

* Wikipedia, *[$C_0$ semigroup](https://en.wikipedia.org/wiki/C0-semigroup)*

[[!redirects one-parameter semigroups]]


one-point compactification

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# Contents
* table of contents
{: toc}

## Idea

The _one-point compactification_ of a [[topological space]] $X$ is a new [[compact space]] $X^* = X \cup \{\infty\}$ obtained by adding a single new point "$\infty$" to the original space and declaring in $X^*$ the [[complements]] of the original [[closed subspace|closed]] [[compact space|compact]] [[subspaces]] to be [[open subspace|open]].

One may think of the new point added as the "point at infinity" of the original space. A [[continuous function]] on $X$ _[[vanishing at infinity|vanishes at infinity]]_ precisely if it extends to a continuous function on $X^*$ and literally takes the value zero at the point "$\infty$".

This one-point compactification is also known as the _Alexandroff compactification_ after a 1924 paper by [[Pavel Aleksandrov|Павел Сергеевич Александров]] (then transliterated 'P.S. Aleksandroff').

The one-point compactification is usually applied to a non-[[compact space|compact]] [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] space.  In the more general situation, it may not really be a [[compactification]] and hence is called the _one-point extension_ or _Alexandroff extension_.

## Definition

### For topological spaces

+-- {: .num_defn #OnePointExtension}
###### Definition
**(one-point extension)**

Let $X$ be any [[topological space]]. Its **one-point extension** $X^*$ is the topological space

* whose underlying [[set]] is the [[disjoint union]] $X \cup \{\infty\}$ of that of $X$ with a [[singleton set]] $\{\infty\}$;

* and whose [[open subsets]] are

  1. the [[open subsets]] of $X \,\subset\, X^*$ ;

  2. the [[complements]] $X^\ast \setminus CK = (X \setminus CK) \cup \{\infty\}$ of the [[closed subspace|closed]] [[compact space|compact]] subsets $CK \subset X$.

=--

([Aleksandrov 24](#Aleksandrov24), see [Kelley 75, p. 150](#Kelley75))

+-- {: .num_remark}
###### Remark

If $X$ is [[Hausdorff space|Hausdorff]], then it is sufficient to speak of compact subsets in def. \ref{OnePointExtension}, since [[compact subspaces of Hausdorff spaces are closed]].

=--

+-- {: .num_lemma #OnePointExtensionWellDefined}
###### Lemma
**(one-point extension is well-defined)**

The [[topological space|topology]] on the one-point extension in def. \ref{OnePointExtension} is indeed well defined in that the given set of subsets is indeed closed under arbitrary unions and finite intersections.

=--

+-- {: .proof}
###### Proof

The unions and finite intersections of the open subsets inherited from $X$ are closed among themselves by the assumption that $X$ is a topological space.

It is hence sufficient to see that

1. the unions and finite intersection of the $(X \backslash CK) \cup \{\infty\}$ are closed among themselves,

1. the union and intersection of a subset of the form $U \underset{\text{open}}{\subset} X \subset X^\ast$ with one of the form $(X \backslash CK) \cup \{\infty\}$ is again of one of the two kinds.

Regarding the first statement: Under [[de Morgan duality]]

$$
  \underset{i \in \underset{\text{finite}}{J}}{\bigcap} (X \backslash CK_i \cup \{\infty\})
  =
  \left( X \backslash \left(\underset{i \in \underset{\text{finite}}{J}}{\bigcup} CK_i \right)\right) \cup \{\infty\}
$$

and

$$
  \underset{i \in I}{\bigcup} ( X \backslash CK_i \cup \{\infty\} )
  =
  \left(X \backslash \left(\underset{i \in I}{\bigcap} CK_i \right)\right) \cup \{\infty\}
$$

and so the first statement follows from the fact that finite unions of compact subspaces and arbitrary intersections of closed compact subspaces are themselves again compact ([this prop.](compact+space#UnionsAndIntersectionOfCompactSubspaces)).

Regarding the second statement: That $U \subset X$ is open means that there exists a closed subset $C \subset X$ with $U = X\backslash C$. Now using [[de Morgan duality]] we find

1. for intersections:

   $$
     \begin{aligned}
       U \cap ( (X\backslash CK) \cup \{\infty\} )
       & =
       (X \backslash C) \cap (X \backslash CK)
       \\
       & =
       X \backslash (C \cup CK).
     \end{aligned}
   $$

   Since finite unions of closed subsets are closed, this is again an open subset of $X$;

2. for unions:

   $$
     \begin{aligned}
       U \cup (X \backslash CK) \cup \{\infty\}
       & =
       (X \backslash C) \cup (X \backslash CK) \cup \{\infty\}
       \\
       & =
       (X \backslash (C \cap CK)) \cup \{\infty\} .
     \end{aligned}
   $$

   For this to be open in $X^\ast$ we need that $C \cap CK$ is again compact. This follows because [[subsets are closed in a closed subspace precisely if they are closed in the ambient space]] and because [[closed subsets of compact spaces are compact]].

=--

### For non-commutative topological spaces ($C^\ast$-algebras)

Dually in [[non-commutative topology]] the one-point compactification corresponds to the [[unitisation of C*-algebras]].

## Properties

### Basic properties
 {#BasicProperties}

We discuss the basic properties of the construction $X^\ast$ in def. \ref{OnePointExtension}, in particular that it always yields a [[compact topological space]] (prop. \ref{OnePointExtensionIsCompact} below) and the ingredients needed to see its [[universal property]] in the Hausdorff case [below](#OnePointExtensionIsCompact).

+-- {: .num_prop #OnePointExtensionIsCompact}
###### Proposition
**(one-point extension is compact)**

For $X$ any [[topological space]], we have that its one-point extension $X^\ast$ (def. \ref{OnePointExtension}) is a [[compact topological space]].

=--

+-- {: .proof}
###### Proof

Let $\{U_i \subset X^\ast\}_{i \in I}$ be an [[open cover]].  We need to show that this has a finite subcover.

That we have a cover means that

1. there must exist $i_\infty \in I$ such that $U_{i \infty} \supset \{\infty\}$ is an [[open neighbourhood]] of the extra point. But since, by construction, the only open subsets containing that point are of the form $(X \backslash CK) \cup \{\infty\}$, it follows that there is a compact closed subset $CK \subset X$ with $X \backslash CK \subset U_{i \infty}$.

1. $\{U_i \subset X\}_{i \in i}$ is in particular an open cover of that closed compact subset $CK \subset X$. This being compact means that there is a finite subset $J \subset I$ so that $\{U_i \subset X\}_{i \in \J \subset X}$ is still a cover of $CK$.

Together this implies that

$$
  \{U_i \subset X\}_{i \in J \subset I}
  \cup
  \{ U_{i_\infty} \}
$$

is a finite subcover of the original cover.

=--

+-- {: .num_prop #HausdorffOnePointCompactification}
###### Proposition
**(one-point extension of locally compact space is Hausdorff precisely if original space is)

Let $X$ be a [[locally compact topological space]]. Then its one-point extension $X^\ast$ (def. \ref{OnePointExtension}) is a [[Hausdorff topological space]] precisely if $X$ is.

=--

+-- {: .proof}
###### Proof

It is clear that if $X$ is not Hausdorff then $X^\ast$ is not.

For the converse, assume that $X$ is Hausdorff.

Since $X^\ast = X \cup \{\infty\}$ as underlying sets, we only need to check that for $x \in X$ any point, then there is an open neighbourhood $U_x \subset X \subset X^\ast$ and an open neighbourhood $V_\infty \subset X^\ast$ of the extra point which are disjoint.

That $X$ is locally compact implies by definition that there exists an open neighbourhood $U_k \supset \{x\}$ whose [[topological closure]] $CK \coloneqq Cl(U_x)$ is a closed compact neighbourhood $CK \supset \{x\}$. Hence

$$
  V_\infty \coloneqq (X \backslash CK ) \cup \{\infty\} \subset X^\ast
$$

is an open neighbourhood of $\{\infty\}$ and the two are disjoint

$$
  U_x \cap V_\infty = \emptyset
$$

by construction.

=--

+-- {: .num_prop #InclusionIntoOnePointExtensionIsOpenEmbedding}
###### Proposition
**(inclusion into one-point extension is open embedding)**

Let $X$ be a [[topological space]]. Then the evident inclusion function

$$
  i \;\colon\; X \longrightarrow X^\ast
$$

into its one-point extension (def. \ref{OnePointExtension}) is

1. a [[continuous function]]

1. an [[open map]]

1. an [[embedding of topological spaces]].

=--

+-- {: .proof}
###### Proof

Regarding the first point: For $U \subset X$ open and $CK \subset X$ closed and compact, the preimages of the corresponding open subsets in $X^\ast$ are

$$
  i^{-1}(U) = U
  \phantom{AAAA}
  i^{-1}( (X \backslash CK) \cup \infty ) = X \backslash CK
$$

which are open in $X$.

Regarding the second point:  The image of an open subset $U \subset X$ is $i(U) = U \subset X^\ast$, which is open by definition.

Regarding the third point: We need to show that $i \colon X \to i(X) \subset X^\ast$ is a [[homeomorphism]]. This is immediate from the definition of $X^\ast$.

=--

+-- {: .num_prop #CompactHausdorffSpaceIsCompactificationOfComplementOfAnyPoint}
###### Proposition

If $X$ is a [[compact Hausdorff space]] and $x_0 \in X$ any point, then $X$ is [[homeomorphism|homeomorphic]] to the one-point compactification (Def. \ref{OnePointExtension}) of its [[complement]] [[subspace]] $X \setminus \{x_0\} \subset X$:

$$
  X \simeq (X \setminus \{x_0\})^\ast
  \,.
$$

Observe also that $X \setminus \{x_0\}$, being an [[open subspace]] of a [[compact Hausdorff space]], is a [[locally compact topological space]], since [[open subspaces of compact Hausdorff spaces are locally compact]], and of course it is Hausdorff, since $X$ is.

=--

+-- {: .proof}
###### Proof

Since [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]], the open neighbourhoods of $x \in X$ are equivalently the
complements of closed, and hence compact closed, subsets in $X \setminus \{x\}$. By def. \ref{OnePointExtension} this means that the function

$$
  \array{
     X &\longrightarrow& (X \setminus \{x_0\})^\ast
  }
$$

which is the identity on $X \setminus \{x_0\}$ and sends $x_0 \mapsto \infty$ (hence which is just the identity on the underlying sets) is a [[homeomorphism]].

=--

### Universal property
  {#UniversalProperty}

As a [[pointed topological space|pointed]] [[locally compact topological space|locally compact]] [[Hausdorff space]], the one-point compactification of $X$ may be described by a [[universal property]]:

For every [[pointed topological space|pointed]] [[locally compact topological space|locally compact]] [[Hausdorff space]] $(Y, y_0)$ and every [[continuous map]] $f \colon X \to Y$ such that the [[pre-image]] $f^{-1}(K)$ is compact for all closed sets $K$ not containing $y_0$, there is a unique basepoint-preserving continuous map $X^\ast \to Y$ that extends $f$.

To see this, note that such a map is necessarily unique. It suffices to show existence. Extend $f$ to a map $f^\ast: X^\ast \to Y$ such that $\infty\mapsto y_0$. If $U\subset Y$ is open, and $y_0\notin U$ then $(f^\ast)^{-1}(U) = f^{-1}(U)$ is open. If $y_0\in U$ then $Y\setminus U$ is closed and $f^{-1}(Y\setminus U)$ is compact, by the hypothesis. Put $K = f^{-1}(Y\setminus U)$; then $K$ is closed and compact and so $(f^\ast)^{-1}(U) = X^\ast\setminus K$ is open. It follows that $f^\ast$ is continuous.

This property characterizes $X^\ast$ in an [[essentially unique]] manner.

$X$ is [[dense subspace|dense]] in $X^*$ precisely if $X$ is not already compact.  Note that $X^*$ is technically a [[compactification]] of $X$ only in this case.

$X^*$ is [[Hausdorff space|Hausdorff]] (hence a [[compactum]]) if and only if $X$ is already both Hausdorff and [[locally compact space|locally compact]] (see prop. \ref{HausdorffOnePointCompactification}).

### Monoidal functoriality
  {#MonoidalFunctoriality}

+-- {: .num_prop #OnePointCompactificationFunctor}
###### Proposition

The operation of one-point compactification (Def. \ref{OnePointExtension}) does not extend to a [[functor]] on the whole [[category]] of [[topological spaces]]. But it does extend to a [[functor]] on [[locally compact Hausdorff spaces]] with [[proper maps]] between them.

=--

(e.g. [Cutler 20, Prop. 1.6](#Cutler20))

+-- {: .num_prop #OnePointCompactificationAndSmashProduct}
###### Proposition
**([[one-point compactification intertwines Cartesian product with smash product]])

On the [[subcategory]] $Top_{LCHaus}$ in [[Top]] of [[locally compact Hausdorff spaces]] with [[proper maps]] between them, the [[functor]] of [[one-point compactification]] (Prop. \ref{OnePointCompactificationFunctor})

$$
  (-)^{cpt}
  \;\colon\;
  Top_{LCHaus}
  \longrightarrow
  Top^{\ast/}
$$

1. sends [[coproducts]], hence [[disjoint union topological spaces]], to [[wedge sums]] of [[pointed topological spaces]];

1. sends [[Cartesian products]], hence [[product topological spaces]], to [[smash products]] of [[pointed topological spaces]];

hence constitutes a [[strong monoidal functor]] for both [[monoidal category|monoidal]] structures of these [[distributive monoidal categories]] in that there are [[natural transformation|natural]] [[homeomorphisms]]

$$
  \big(
    X \sqcup Y
  \big)^{cpt}
  \;\simeq\;
  X^{cpt} \vee Y^{cpt}
  \,,
$$

and

$$
  \big(
    X \times Y
  \big)^{cpt}
  \;\simeq\;
  X^{cpt} \wedge Y^{cpt}
  \,.
$$

=--

This is briefly mentioned in [Bredon 93, p. 199](#Bredon93).
The argument is spelled out in: [MO:a/1645794](https://math.stackexchange.com/a/1645794/58526), [Cutler 20, Prop. 1.6](#Cutler20).

## Examples

### General

\begin{example}
\label{OnePointExtensionOfCompactSpace}
  If $X$ is already itself [[compact topological space|compact]], then its one-point extension in the sense of Def. \ref{OnePointExtension} is the [[disjoint union]] of $X$ with a [[singleton]] $\{ \infty \}$.

Namely, in this case the open neighbourhood $(X \setminus X) \cup \{\infty\}$ of $\infty$ consists of just $\infty$ itself, which is hence an [[open point]]. But it is also a [[closed point]], being the [[complement]] of $X$.
\end{example}

### Compactification of discrete spaces

\begin{example}\label{CompactifOfCountableSet}
**(one-point compactification of [[countable set|countable]] [[discrete space]])**
\linebreak
Consider the [[natural numbers]] $\mathbb{N}$ regarded as a [[discrete space]]. This is not [[compact topological space|compact]]. Its one-point compactification has as [[underlying set]] the [[disjoint union]]
$\mathbb{N}\cup\{\infty\}$ of the natural numbers with an element "at [[infinity]]", and its [[open subsets]] are those [[subsets]] that either do not contain $\infty$, or which contain $\infty$ and are [[cofinite subsets]].

This space is actually a [[Stone space]], and corresponds via [[Stone duality]] to the [[Boolean algebra]] of the finite and cofinite subsets of $\mathbb{N}$, with the usual [[Boolean algebra]] operations of [[union]] and [[complement|set-complement]].

To see this, notice that the [[clopen sets]] in the space are those that are either [[finite set|finite]] and not containing $\infty$, or [[cofinite set|cofinite]] and containing $\infty$. So a clopen set is determined by giving either a finite or a cofinite subset of $\mathbb{N}$, and then adding $\infty$ if it is cofinite. Under this correspondence, the Boolean algebra operations on finite and cofinite subsets of $\mathbb{N}$ correspond to the Boolean algebra operations on the clopen sets.

Another reason that this space is important is because to give a continuous map $\mathbb{N}\cup\{\infty\}\to X$ to a topological space $X$ is to give a convergent sequence in $X$. This can then be used as a foundation: [[Johnstone's topological topos]] is a category of [[sheaves]] on the continuous [[endofunctions]] of $(\mathbb{N}\cup\{\infty\})$, and [[subsequential spaces]] are a subcategory of [[concrete sheaves]].
\end{example}

### Euclidean spaces compactify to Spheres
 {#ExamplesSpheres}

We discuss how the [[one-point compactification]] of [[Euclidean space]] of [[dimension]] $n$ is the [[n-sphere]].

+-- {: .num_example #nSphereIsOnePointCompactificationOfRn}
###### Example
**([[one-point compactification]] of [[Euclidean space|Euclidean n-space]] is the [[n-sphere]])

For $n \in \mathbb{N}$ the [[n-sphere]] with its standard [[topological space|topology]] (e.g. as a [[subspace]] of the [[Euclidean space]] $\mathbb{R}^{n+1}$ with its [[metric topology]]) is [[homeomorphism|homeomorphic]] to the one-point compactification (def. \ref{OnePointExtension}) of the [[Euclidean space]] $\mathbb{R}^n$

$$
  S^n \simeq (\mathbb{R}^n)^\ast
  \,.
$$

=--

+-- {: .proof}
###### Proof

Pick a point $\infty \in S^n$. By [[stereographic projection]] we have a [[homeomorphism]]

$$
  S^n \setminus \{\infty\} \simeq \mathbb{R}^n
  \,.
$$

With this it only remains to see that for $U_\infty \supset \{\infty\}$ an open neighbourhood of $\infty$ in $S^n$ then the complement $S^n \setminus U_\infty$ is compact closed, and conversely that the complement of every compact closed subset of $S^n \setminus \{\infty\}$ is an open neighbourhood of $\{\infty\}$.

Observe that under [[stereographic projection]] the open subspaces $U_\infty \setminus \{\infty\} \subset S^n \setminus \{\infty\}$ are identified precisely with the [[closed subset|closed]] and [[bounded subsets]] of $\mathbb{R}^n$. (Closure is immediate, boundedness follows because an open neighbourhood of $\{\infty\} \in S^n$ needs to contain an open ball around $0 \in \mathbb{R}^n \simeq S^n \setminus \{-\infty\}$ in the _other_ stereographic projection, which under change of chart gives a bounded subset. )

By the [[Heine-Borel theorem]] the closed and bounded subsets of $\mathbb{R}^n$ are precisely the compact, and hence the compact closed, subsets of $\mathbb{R}^n \simeq S^n \setminus \{\infty\}$.

=--

+-- {: .num_remark #RelevanceForMonopolesAndInstantons}
###### Remark
**(relevance for [[monopoles]] and [[instantons]] in [[gauge theory]])**

In [[physics]], Example \ref{nSphereIsOnePointCompactificationOfRn} governs the phenomenon of [[monopoles]] and [[instantons]] for [[gauge theory]] on [[Minkowski spacetime]] or [[Euclidean space]]: While such spaces themselves are not [[compact topological space|compact]], the consistency condition that any [[field history|field configuration]] carries a [[finite number|finite]] [[energy]] requires that [[gauge fields]] _[[vanishing at infinity|vanish at infinity]]_.

This means that if $A$ is the [[classifying space]] for the corresponding gauge field --  e.g. $A = B G$ for [[Yang-Mills theory]] with [[gauge group]] $G$ -- and if $a_\infty \in A$ denotes the [[pointed topological space|base point]] witnessing vanishing fields, then a [[field configuration]]/[[cocycle]]

$$
  \mathbb{R}^n
  \overset{c}{\longrightarrow}
  A
$$

on $\mathbb{R}^n$ in $A$-[[generalized cohomology theory|cohomology]] _[[vanishing at infinity|vanishes at infinity]]_ if outside any [[compact topological space|compact]] [[subset]] its value is the vanishing field configuration $a_\infty$.

But by Def. \ref{OnePointExtension} this is equivalent to the cocycle [[extension|extends]] to the [[one-point compactification]] as a [[morphism]] of [[pointed topological space]]:

$$
  \array{
    \big(
      \mathbb{R}^n
    \big)
    \simeq S^n
    &
    \overset{c}{\longrightarrow}
    &
    A
    \\
    \infty &\mapsto& a_\infty
  }
$$

The following graphics illustrates this for $A = S^n$ an [[n-sphere]] itself, hence for charges in [[Cohomotopy cohomology theory]]:

> graphics grabbed from [SS 19](cohomotopy+charge#SatiSchreiber19)

For more see at _[Yang-Mills instanton -- SU(2)-instantons from the correct maths to the traditional physics story](Yang-Mills+instanton#FromTheMathsToThePhysicsStory)_.

=--

### Linear representations compactify to representation spheres

Via the presentation of example \ref{nSphereIsOnePointCompactificationOfRn}, the canonical [[action]] of the [[orthogonal group]] $O(N)$ on $\mathbb{R}^n$ induces an action of $O(n)$ on $S^n$, which preserves the basepoint $\infty$ (the "point at infinity").

This construction presents the [[J-homomorphism]] in [[stable homotopy theory]] and is encoded for instance in the definition of [[orthogonal spectra]].

Slightly more generally, for $V$ any real [[vector space]] of [[dimension]] $n$ one has $S^n \simeq (V)^\ast$. In this context and in view of the previous case, one usually writes

$$
  S^V \coloneqq (V)^\ast
$$

for the $n$-[[sphere]] obtained as the one-point compactification of the vector space $V$.

As a special case of Prop. \ref{OnePointCompactificationAndSmashProduct} we have:

+-- {: .num_prop }
###### Proposition

For $V,W \in Vect_{\mathbb{R}}$ two real [[vector spaces]], there is a [[natural transformation|natural]] [[homeomorphism]]

$$
  S^V \wedge S^W \simeq S^{V\oplus W}
$$

between the [[smash product]] of their one-point compactifications and the one-point compactification of the [[direct sum]].

=--

+-- {: .num_remark }
###### Remark

In particular, it follows directly from this that the [[suspension]] $\Sigma(-) \simeq S^1 \wedge (-)$ of the $n$-sphere is the $(n+1)$-sphere, up to [[homeomorphism]]:

$$
  \begin{aligned}
    \Sigma S^n
    & \simeq S^{\mathbb{R}^1} \wedge S^{\mathbb{R}^n}
    \\
    & \simeq S^{\mathbb{R}^1 \oplus \mathbb{R}^n}
    \\
    & \simeq S^{\mathbb{R}^{n+1}}
    \\
    & \simeq S^{n+1}
  \end{aligned}
  \,.
$$

=--

\linebreak

### Thom spaces

For $X$ a [[compact topological space]] and $V \to X$
a [[vector bundle]], then the ([[homotopy type]] of the) one-point compactification of
the total space $V$ is the [[Thom space]] of $V$, equivalent to $D(V)/S(V)$.

For a simple example: the real [[projective plane]] $\mathbb{RP}^2$ is the one-point compactification of the 'open' [[Moebius strip|Möbius strip]], as line bundle over $S^1$. This is a special case of the more general observation that $\mathbb{RP}^{n+1}$ is the Thom space of the [[tautological line bundle]] over $\mathbb{RP}^n$.

### Locally compact Hausdorff spaces

+-- {: .num_example #LocallyCompatcHausdorffSpaceIsOpenSubspaceOfCompactHausdorffSpace}
###### Example
**(every [[locally compact topological space|locally compact]] [[Hausdorff space]] is an [[open subset|open]] [[subspace]] of a [[compact Hausdorff space]])**

Every [[locally compact topological space|locally compact]] [[Hausdorff space]]  is [[homeomorphism|homeomorphic]] to a [[open subset|open]] [[topological subspace]] of a [[compact topological space]].

=--

+-- {: .proof}
###### Proof

In one direction the statement is that [[open subspaces of compact Hausdorff spaces are locally compact]] (see there for the proof). What we need to show is that every locally compact Hausdorff spaces arises this way.

So let $X$ be a locally compact Hausdorff space. By prop. \ref{OnePointExtensionIsCompact} and prop. \ref{HausdorffOnePointCompactification} its one-point extension $X^\ast$ (def. \ref{OnePointExtension}) is a [[compact Hausdorff space]]. By prop. \ref{InclusionIntoOnePointExtensionIsOpenEmbedding} the canonical inclusion $X \to X^\ast$ is an [[open map|open]] [[embedding of topological spaces]].

=--

## Related concepts

* [[Thom space]]

* [[compactification]]

  * [[Stone-Cech compactification]]

  * [[end compactification]]

  * [[Bohr compactification]]

  * [[Kaluza-Klein compactification]]

* [[configuration space (mathematics)]]

* [Yang-Mills instanton -- SU(2)-instantons from the correct maths to the traditional physics story](Yang-Mills+instanton#FromTheMathsToThePhysicsStory)_

## References

The concept goes back to

* {#Aleksandrov24} [[Pavel Aleksandrov]], _Über die Metrisation der im Kleinen kompakten topologischen Räume_, Mathematische Annalen (1924) Volume: 92, page 294-301 ([dml:159072](https://eudml.org/doc/159072))

Textbook accounts:

* {#Kelley75} [[John Kelley]], p. 150 of: *General topology*, D. van Nostrand, New York (1955), reprinted as: Graduate Texts in Mathematics, Springer (1975) [[ISBN:978-0-387-90125-1](https://www.springer.com/gp/book/9780387901251)]

* {#Bredon93} [[Glen Bredon]], p. 199 of: _Topology and Geometry_, Graduate Texts in Mathematics 139, Springer 1993 ([doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0), [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf))

Review:

* {#Cutler20} [[Tyrone Cutler]], _The category of pointed topological spaces_ (2020) [[pdf](https://www.math.uni-bielefeld.de/~tcutler/pdf/Elementary%20Homotopy%20Theory%20II%20-%20The%20Pointed%20Category.pdf), [[CutlerPointedTopologicalSpaces.pdf:file]]]

See also

* Wikipedia, _[Alexandroff extension](https://en.wikipedia.org/wiki/Alexandroff_extension)_

[[!redirects one-point compactification]]
[[!redirects one-point compactifications]]
[[!redirects One-point compactification]]
[[!redirects Alexandroff compactification]]
[[!redirects Alexandroff compactifications]]
[[!redirects Alexandrov compactification]]
[[!redirects Alexandrov compactifications]]
[[!redirects Aleksandrov compactification]]
[[!redirects Aleksandrov compactifications]]
[[!redirects Александров compactification]]
[[!redirects Александров compactifications]]

[[!redirects one-point extension]]
[[!redirects one-point extensions]]
[[!redirects Alexandroff extension]]
[[!redirects Alexandroff extensions]]
[[!redirects Alexandrov extension]]
[[!redirects Alexandrov extensions]]
[[!redirects Aleksandrov extension]]
[[!redirects Aleksandrov extensions]]
[[!redirects Александров extension]]
[[!redirects Александров extensions]]

[[!redirects point at infinity]]


one-point compactification intertwines Cartesian product with smash product

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_prop #OnePointCompactificationAndSmashProduct}
###### Proposition
**([[one-point compactification]] of [[product topological space|product space]] is [[smash product]] of the compactified factors)

On the [[subcategory]] $Top_{LCHaus}$ in [[Top]] of [[locally compact Hausdorff spaces]] with [[proper maps]] between them, the [[functor]] of [[one-point compactification]] (Prop. \ref{OnePointCompactificationFunctor})

$$
  (-)^{cpt}
  \;\colon\;
  Top_{LCHaus}
  \longrightarrow
  Top^{\ast/}
$$

1. sends [[coproducts]], hence [[disjoint union topological spaces]], to [[wedge sums]] of [[pointed topological spaces]];

1. sends [[Cartesian products]], hence [[product topological spaces]], to [[smash products]] of [[pointed topological spaces]];

hence constitutes a [[strong monoidal functor]] for both [[monoidal category|monoidal]] structures of these [[distributive monoidal categories]] in that there are [[natural transformation|natural]] [[homeomorphisms]]

$$
  \big(
    X \sqcup Y
  \big)^{cpt}
  \;\simeq\;
  X^{cpt} \vee Y^{cpt}
  \,,
$$

and

$$
  \big(
    X \times Y
  \big)^{cpt}
  \;\simeq\;
  X^{cpt} \wedge Y^{cpt}
  \,.
$$

=--

This is briefly mentioned in [Bredon 93, p. 199](#Bredon93).
The argument is spelled out in: [MO:a/1645794](https://math.stackexchange.com/a/1645794/58526), [Cutler 20, Prop. 1.6](#Cutler20).

## References

Basic accounts:

* {#Bredon93} [[Glen Bredon]], p. 199 of: _Topology and Geometry_, Graduate Texts in Mathematics 139, Springer 1993 ([doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0), [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf))

Review:

* {#Cutler20} [[Tyrone Cutler]], _The category of pointed topological spaces_ (2020) [[pdf](https://www.math.uni-bielefeld.de/~tcutler/pdf/Elementary%20Homotopy%20Theory%20II%20-%20The%20Pointed%20Category.pdf), [[CutlerPointedTopologicalSpaces.pdf:file]]]


one-sided real number

# One-sided real numbers
* table of contents
{: toc}

## Summary

There are really two notions of one-sided real number: lower reals and upper reals.  But there is a nearly perfect symmetry between them (although the lower reals seem to get slightly more use).

These are largely concepts in [[constructive mathematics]].  Using [[excluded middle]], a bounded one-sided real number is simply a [[real number]], but constructively they are more general (and less well behaved).  Lower reals are particularly important in constructive [[measure theory]] as the values taken by [[positive measures]] (and more generally are used to define [[extended positive cones]]).

However, even in [[classical mathematics]], there are some differences, at least in emphasis and application.  First, it is most natural to include $\infty$ as a lower real and include $-\infty$ as an upper real.  (However, you can restrict to _bounded_ lower/upper reals to avoid that, or alternatively genralise to _extended_ lower/upper reals if you want to include both $\pm\infty$ at once.)  Also, the natural [[topological structure|topology]] on the lower reals is the _lower [[semicontinuous topology]]_ (and the natural topology on the upper reals is the _upper semicontinuous topology_).

One-sided numbers are sometimes called (upper or lower) *semicontinuous* numbers.  To see why, consider a [[locale]] (or [[topological space]]) $L$, and consider the (upper or lower) [[semicontinuous functions]] (real-valued) on $L$.  On the one hand, these are the same as the [[continuous functions]] from $L$ to (classically) the set of real numbers equipped with the semicontinuous topology or to (constructively) a locale whose [[point of a locale|points]] are the one-sided real numbers.  On the other hand, these semicontinuous functions are precisely the [[internal logic|internal]] one-sided real numbers in the [[topos of sheaves]] over $L$.

Lower and upper reals don\'t interact well together.  Their "intersection" is the set of [[MacNeille real numbers]], which is moderately well-behaved; it also contains the set of [[located real numbers]], which is better-behaved algebraically (but not, constructively, a complete lattice).  But it seems hard, constructively, to find any reasonably behaved notion of "real number" than contains both upper and lower reals.

## Idea

(In the following, 'number' without qualification may be taken to mean either a [[real number]] or a [[rational number]]; the result is the same either way.  Indeed, we could take any [[dense subset|dense set]] of real numbers.)

A __lower real number__ is the [[supremum]] of an [[inhabited set|inhabited]] set of numbers.  A __bounded lower real__ is the supremum of an inhabited set of numbers that has a finite upper bound.  An __extended lower real__ is the supremum of an arbitrary set of numbers.

An __upper real number__ is the [[infimum]] of an inhabited set of numbers.  A __bounded upper real__ is the infimum of an inhabited set of numbers that has a finite lower bound.  An __extended lower real__ is the infimum of an arbitrary set of numbers.

We cannot generalize further by taking more extrema of the same sort.  Explicitly, the supremum of any inhabited set of lower reals is a lower real and the infimum of any inhabited set of upper reals is an upper real.  Similar results obtain if we use either bounded or extended reals, so long as we also either require the set to bounded or allow it to be [[empty subset|empty]].

## Definitions

There are some choices as to how exactly to represent one-sided real numbers; we will define them here as certain [[subsets]] of the set $\mathbb{Q}$ of [[rational numbers]].  If we replaced $\mathbb{Q}$ with any other [[dense subset]] of the [[real line]], then the definitions would end up being equivalent.

### Lower reals

An __extended lower real number__ is any subset $L$ of $\mathbb{Q}$ such that

*  if $a \in L$, then $a \lt b$ for some $b \in L$; and
*  if $a \lt b$ and $b \in L$, then $a \in L$.

We can actually combine these into a single rule:

*  $a \in L$ if and only if $a \lt b$ for some $b \in L$.

A __lower real number__ is an extended lower real $L$ such that:

*  some $a \in L$.

A __bounded lower real number__ is a lower real $L$ such that

*  some $b \notin L$.

When treated explicitly as a subset of $\mathbb{Q}$, we call $L$ a __[[lower set]]__.  When we interpret $L$ as a one-sided real number $x$, we write $a \lt x$ to mean that $a \in L$.  Conversely, we can interpret any rational number, or even any real number, $x$ as a bounded lower real, specifically as the set of all rational numbers less than $x$.  (A bounded lower real is __located__ if it is the lower real obtained in this way from a real number; classically, every bounded lower real number is located.)  We interpret $\infty$ as the lower real whose lower set is all of $\mathbb{Q}$; we interpret $-\infty$ as the extended lower real whose lower set is empty.

### Upper reals

An __extended upper real number__ is any subset $U$ of $\mathbb{Q}$ such that

*  if $b \in U$, then $a \lt b$ for some $a \in U$; and
*  if $a \lt b$ and $a \in U$, then $b \in U$.

We can actually combine these into a single rule:

*  $b \in U$ if and only if $a \lt b$ for some $a \in U$.

An __upper real number__ is an extended upper real $U$ such that:

*  some $b \in U$.

A __bounded upper real number__ is an upper real $U$ such that

*  some $a \notin U$.

When treated explicitly as a subset of $\mathbb{Q}$, we call $U$ an __[[upper set]]__.  When we interpret $U$ as a one-sided real number $x$, we write $x \lt b$ to mean that $b \in U$.  Conversely, we can interpret any rational number, or even any real number, $x$ as a bounded upper real, specifically as the set of all rational numbers greater than $x$.  (A bounded upper real is __located__ if it is the upper real obtained in this way from a real number; classically, every bounded upper real number is located.)  We interpret $-\infty$ as the upper real whose upper set is all of $\mathbb{Q}$; we interpret $\infty$ as the extended upper real whose upper set is empty.

## Order relations

We define [[order]] relations between extended lower reals $x$ and $y$ as follows:

*  $x \lt y$ if and only if there exists a rational number $a$ such that $a \lt y$ but not $a \lt x$;
*  $x \leq y$ if and only if $a \lt y$ for every rational number $a$ such that $a \lt x$.

In other words, $\leq$ is $\subseteq$ on lower sets, and $\lt$ is $\subsetneq$ (in an appropriate sense).

We define order relations between extended upper reals $x$ and $y$ as follows:

*  $x \lt y$ if and only if there exists a rational number $b$ such that $x \lt b$ but not $y \lt b$;
*  $x \leq y$ if and only if $x \lt b$ for every rational number $b$ such that $y \lt b$.

In other words, $\leq$ is $\supseteq$ on upper sets, and $\lt$ is $\supsetneq$.

We now have multiple meanings for $x \lt y$ or $x \leq y$ if one or both of these is already a rational number or a real number, but it is a theorem that the meanings are all consistent.  We also have $-\infty \leq x \leq \infty$ for every extended number $x$, and $-\infty \lt x \lt \infty$ for every bounded number $x$.  Finally, $x \leq y$ if $x \lt y$ or $x = y$; the converse holds classically.

Each version of $\leq$ is a [[partial order]] and each version of $\lt$ is a [[quasiorder]].  By [[excluded middle]], $\leq$ is a [[total order]] and $\lt$ is the corresponding [[strict total order]], but neither of these results holds constructively.  In particular, the [[comparison]] law for $\lt$ is invalid, which is why one-sided reals are not as well behaved as ordinary (located) [[real numbers]].

## Suprema and infima

Given any collection of extended lower reals, their __supremum__ is an extended lower real which is found by taking the [[union]] of lower sets.  The supremum of an inhabited set of lower reals is a lower real, and the supremum of an inhabited set of bounded lower reals is bounded iff the set has a bounded lower real as an upper bound.  In any case, this supremum really is the [[supremum]] under $\leq$.  This agrees with the notion of supremum of real numbers in the real number system, when that exists.  Every extended lower real is also the supremum in this sense of its corresponding lower set, interpreted as a set of lower reals that happen to all be rational numbers.

Given any collection of extended upper reals, their __infimum__ is an extended upper real which is found by taking the union of upper sets.  The infimum of an inhabited set of upper reals is an upper real, and the infimum of an inhabited set of bounded upper reals is bounded iff the set has a bounded upper real as a lower bound.  In any case, this infimum really is the [[infimum]] under $\leq$.  This agrees with the notion of infimum of real numbers in the real number system, when that exists.  Every extended upper real is also the infimum in this sense of its corresponding upper set, interpreted as a set of upper reals that happen to all be rational numbers.

By the [[adjoint functor theorem]], suprema of upper reals and infima of lower reals also exist, but they are not constructed as set-theoretic intersections but rather as the [[interior]] of an intersection.  (This construction makes sense even in [[predicative mathematics|predicative]] constructive mathematics, though the adjoint functor theorem does not.)  However, these may not be what we really want; an obvious example of this is if we take the infimum of a set of lower reals that happen to all be located, in which case what we really want is the corresponding *upper* real infimum (for instance, when defining [[located subspaces]]).  (In the [[MacNeille real numbers]], contained in both the lower and upper reals, both infima and suprema are constructed using interiors, and hence neither may be what we want.)

An [[inferior limit]] is ...

## Arithmetic {#arithmetic}

In general, you can extend [[order-preserving functions]] of rational numbers to lower reals and to upper reals; you simply take the [[image]] of the lower or upper set and close it downward or upwards, as appropriate.  If you have an order-reversing function of rational numbers, then you can apply it to a lower real to get an upper real and conversely; this is about the only interaction that I know of between the two kinds of one-sided real.

Since addition is order-preserving, it works nicely, at least to make a [[monoid]]; but [[subtraction]] doesn\'t work in general.  Multiplication has trouble with negative numbers but produces good results if you restrict to $[0,\infty]$.  Notice that we get $0 \cdot \infty = 0$ with lower reals but $0 \cdot \infty = \infty$ with extended upper reals; thus, ${[{0,\infty}[}$ is a [[rig]] for either lower or upper reals, while $[0,\infty]$ is a rig only for lower reals.  (This doesn\'t so much mean that upper reals behave worse than lower reals, as that extended reals on either side behave worse.)  We can also use [[logarithms]] to translate between addition and multiplication.  (In particular, $-\infty + \infty = -\infty$ with extended lower reals but $-\infty + \infty = \infty$ with extended upper reals.)

Of course, arithmetic with one-sided real numbers is much easier than this in [[classical mathematics]], where the bounded or extended versions may be identified; but even there, the natural operations involving $\pm\infty$ may differ.  Accordingly, the conventions for arithmetic with infinities can signal which sort of number is appropriate for a [[constructive mathematics|constructive]] development.

If you want to do arbitrary arithmetic operations constructively on something more general than [[located real numbers]], you can try the [[MacNeille real number|MacNeille reals]], which are contained in both the upper and the lower reals.

## Topology

Let\'s put this on a separate page: [[semicontinuous topology]].  For one thing, it\'s more advanced than the elementary stuff above; for another thing, it\'s more interesting in classical mathematics; and finally, it\'s more complicated in constructive mathematics.  The main point is that each type of one-sided real number has its own natural topology, and these differ (both classically and constructively) from the usual topology on the [[real line]].

## In topos theory

There is a bijective correspondence of [[internalization|internal]] extended lower real numbers in a sheaf topos $\mathrm{Sh}(X)$ of a [[topological space]] $X$ and [[semicontinuous map|lower semicontinuous]] functions $X \to \mathbb{R} \cup \{ +\infty \}$, or equivalently a [[continuous map|continuous]] function $X \to \mathbb{R} \cup \{ +\infty \}$, when the latter is equipped with the [[lower semicontinuous topology]].  (We will work classically in the external logic.)

Let $\Delta(\mathbb{Q})$ denote the [[constant sheaf]] of rational numbers on $X$. A _lower real number_ $U$ in $\mathrm{Sh}(X)$ is then defined as a subsheaf of $\Delta(\mathbb{Q})$ satisfying the axioms above (interpreted in the [[internal language]]). Such a lower real number induces a lower semicontinuous function $x \mapsto \sup U_x$, where $U_x$ denotes the [[stalk]] of $U$ at $x$ (taken as a subset of $\mathbb{Q}$).

Conversely, a lower semicontinuous function $f : X \to \mathbb{R} \cup \{ +\infty \}$ defines an internal lower real number $U$ with sections
$$U(A) \coloneqq \{ \varphi : A \to \mathbb{Q} | \forall x \in A: \varphi(x) \lt f(x) \} \subseteq \Delta(\mathbb{Q})(A)$$
over open sets $A \subseteq X$.

See [Mulvey (1974, page 28)](#Mulvey1974) for details.

For the externally constructive version of this, it is best to take $X$ to be a [[locale]]; we need to use the (external) locale $L$ of lower real numbers in place of $\mathbb{R} \cup \{\infty\}$, and the notion of semicontinuous map is replaced with a continuous map (in the sense of locales) to $L$.

## References

* D. Lesnik, _Synthetic Topology and Constructive
Metric Spaces_ , PhD University of Ljubljana 2010. ([pdf](http://www.fmf.uni-lj.si/storage/19160/PhD_Davorin.pdf))

* [[Chris Mulvey|C. Mulvey]] (1974): _Intuitionistic Algebra and Representations of Rings_. In Hofmann, Liukkonen: _Recent Advances in the Representation Theory of Rings and C\*-Algebras by Continuous Sections_, AMS 1974.
  {#Mulvey1974}

* J. Z. Reichman, _Semicontinuous Real Numbers in a Topos_ , JPAA **28** (1983) pp.81-91.

* Guillaume Raynaud, *Fibred Contextual Quantum Physics*, PhD thesis, Birmingham  (2014) [[etheses:1685](https://etheses.bham.ac.uk/id/eprint/1685/), [pdf](https://etheses.bham.ac.uk//id/eprint/1685/1/Raynaud14PhD.pdf)]

See also the following MO-discussion: ([link](http://mathoverflow.net/questions/108029/simplification-in-semi-continuous-real?rq=1))

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one-to-one correspondence

\tableofcontents

## Definition

### In set theory

In [[set theory]], there are many ways to say that given the [[relation]] or [[correspondence]] $R$, $R(x, y)$ holds of elements $x \in A$ and $y \in B$. In [[allegorical set theories]] like [[SEAR]], it is a foundational concept. In [[categorical set theories]] like [[ETCS with elements]], we say that $R(x, y)$ holds of elements $x \in A$ and $y \in B$ if the ordered pair $(x, y)$ is in the [[image]] of the function $(p_1, p_2):R \to A \times B$. In [[material set theories]] like [[ZFC]], we say that $R(x, y)$ holds of elements $x \in A$ and $y \in B$ if the ordered pair $(x, y)$ is in $R$.

In [[set theory]], a **one-to-one correspondence** is a [[relation]] or [[correspondence]] $R$ between sets $A$ and $B$ for which for every element $x \in A$, there is a unique element $y \in B$ such that $R(x, y)$ holds, and for every element $y \in B$, there is a unique element $x \in A$ such that $R(x, y)$ holds. Equivalently, it is an [[anafunction]] $R$ between sets $A$ and $B$ such that for every element $y \in B$, there is a unique element $x \in A$ such that $R(x, y)$ holds.

### In type theory

In [[type theory]], given types $A$ and $B$, a **one-to-one correspondence** is a [[correspondence]] $x:A, y:B \vdash R(x, y)$ for which for every element $x:A$, [[uniqueness quantifier|there exists a unique]] element $y:B$ such that $R(x, y)$, and for every element $y:B$, there exists a unique element $x:A$ such that $R(x, y)$. Equivalently, it is an [[anafunction]] $x:A, y:B \vdash R(x, y)$ such that for every element $y:B$, the dependent type $\sum_{x:A} R(x, y)$ is a [[contractible type]].

Given a [[Tarski universe]] $(U, T)$, the type of all $U$-small one-to-one correspondences between $A$ and $B$ is given by the type

$$\mathrm{OneToOneCorr}_U(A, B) \coloneqq \sum_{R:A \times B \to U} \left(\prod_{x:A} \exists!y:B.T(R(x, y)\right) \times \left(\prod_{y:B} \exists!x:A.T(R(x, y))\right)$$

The univalence axiom for $U$ is given by the [[equivalence of types]]

$$\mathrm{ua}_U:\prod_{A:U} \prod_{B:U} (A =_U B) \simeq \mathrm{OneToOneCorr}_U(T(A), T(B))$$

## Properties

By the [[principle of unique choice]], given every one-to-one correspondence, there is a [[bijection]] $f:A \cong B$. However, in the absence of the principle of unique choice, bijections and one-to-one correspondences are not necessarily the same; this is the same phenomena as the phenomena between [[functions]] and [[anafunctions]]. However, frequently in [[material set theories]], functions are defined to be anafunctions, and so bijections are similarly defined to be one-to-one correspondences.

Something similar occurs in [[dependent type theory]], where by the [[principle of unique choice]], given every one-to-one correspondence, there is a function with contractible fibers $x:A \vdash f(x):B$, which becomes a bijection in the context of [[axiom K]] or [[uniqueness of identity proofs]]. However, in contrast to [[material set theory]], in dependent type theory, there is always a distinction between a function $x:A \vdash f(x):B$ and an anafunction $x:A, y:B \vdash R(x, y)$, and so there is always a distinction between a function with contractible fibers and a one-to-one correspondence.

## See also

* [[bijection]], [[equivalence of types]]

* [[anafunction]]

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Online > history

< [[Online]]


Onsager-Machlup function

#Onsager-Machlup
The Onsager-Machlup function is a generalization of a density (Radon-Nikodym derivative) to non locally compact spaces. Rather than comparing a probability measure to a translation invariant measure, the idea is to compare a probability measure to translations of itself.

More precisely, consider a locally compact metric group $(G, d)$ with Borel probability measure $\mu$.  Then there exists Haar measure $\lambda$, and we may thus consider the limit
\[f(z):=\lim_{\varepsilon\to 0} \frac{\mu(B_\varepsilon(z))}{\lambda(B_\varepsilon(z))}.\]

If this limit exists, then $f(z)$ is called the Radon-Nikodym derivative or density of $\mu$. However if $G$ is not locally compact (for instance, an infinite dimensional Banach space), then there is not a Haar measure. Therefore, we may consider the limit
\[f(z):=\lim_{\varepsilon\to 0} \frac{\mu(B_\varepsilon(z))}{\mu(B_\varepsilon(0))}.\]

If this limit exists, then $f(z)$ can be viewed as a fictitious density of the measure $\mu$. By convention, we call $\operatorname{OM}_\mu(z)=- \log (f(z))$ the Onsager-Machlup function for $\mu$.

#Mode
The Onsager-Machlup is the minimization objective for the mode of the measure $\mu$. That is, by minimizing $\operatorname{OM}_\mu(z)$, we maximize $f(z)$ which yields the modes of $\mu$.

#Example
The Onsager-Machlup function for an infinite dimensional centered Gaussian measure $\mu$ on Banach space $B$ with Cameron-Martin space $H_\mu$ is
\[\operatorname{OM}_\mu(z)=
\frac{1}{2}\|z\|_{\mu}^2 .\]

if $z\in H_\mu$ and $\infty$ otherwise.


onto morphism in a dagger 2-poset

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## Contents ##
* table of contents
{:toc}

## Idea ##

A morphism similar to how onto relations are defined in [[Rel]]

## Definition ##

A morphism $f \in hom_A(a,b)$ of a [[dagger 2-poset]] $A$ is an **onto morphism** if $1_b \leq f \circ f^\dagger$.

## See also ##

* [[functional morphism in a dagger 2-poset]]
* [[entire morphism in a dagger 2-poset]]
* [[injective morphism in a dagger 2-poset]]


ontology

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## Idea

Ontology is the branch of [[philosophy]] concerned with [[being]] and [[becoming]].

## Related concepts

* [[metaphysics]]

* [[category of being]]

* [[first law of thought]]

## References

* Wikipedia, [Ontology](http://en.wikipedia.org/wiki/Ontology)

* [[Georg Hegel]], _[[Science of Logic]]_

* [[Ludwig Wittgenstein]], _[[Tractatus Logico-Philosophicus]]_

[[!redirects ontologies]]


op modality

\tableofcontents

## Idea

In [[simplicial type theory]] and [[bicubical type theory]] and related approaches to [[synthetic (infinity,1)-category theory]], the **op modality** is a [[modality]] $(-)^\op$ which takes a [[Segal type]], representing (infinity,1)-precategories, and turns it into the opposite [[Segal type]], representing the opposite (infinity,1)-precategory.

## Related concepts

* [[Segal type]]

* [[Rezk type]]

## References

* [[Jonathan Weinberger]], *Type-theoretic modalities for synthetic $(\infty, 1)$-categories*, Talk at [Homotopy Type Theory 2019](https://hott.github.io/HoTT-2019/), August 14, 2019. ([slides](https://hott.github.io/HoTT-2019//conf-slides/Weinberger.pdf))


open cover

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## Definition

An **open cover** of a [[topological space]] $X$ is a collection $\{U_i \subset X\}$ of [[open subset]]s of $X$ whose [[union]] equals $X$: $\cup_i U_i = X$.

## Properties

When denoting by $U_i \hookrightarrow X$ the [[inclusion morphisms]] in the [[category]] [[Top]], each open cover constitutes a [[covering]] family $\{U_i \to X\}$ in the sense of [[sheaf and topos theory]] which is a standard [[coverage]] on [[Top]].

Analogous statements hold for categories of topological spaces with extra [[stuff, structure, property|structure]], such as the category [[Diff]] of [[smooth manifold]]s.

If an open cover has the property that all the $U_i$ and all of their finite nonempty intersections are [[contractible]], then one speaks of a _[[good open cover]]_.

## Related concepts

* [[compact space]]

* [[finite open cover]]

  [[locally finite open cover]]

  [[countable open cover]]

  [[numerable open cover]]

  [[good open cover]]

* [[equivariant open cover]]

* [[closed cover]]

* [[schedule]]

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open covers of metric spaces have open countably locally discrete refinements

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## Definitions

+-- {: .num_defn #LocallyFiniteCover}
###### Definition
**([[locally finite cover]])**

Let $(X,\tau)$ be a [[topological space]].

An [[open cover]] $\{U_i \subset X\}_{i \in I}$ of $X$ is called  _locally finite_ if for each point $x \in X$, there exists a  [[neighbourhood]] $U_x \supset \{x\}$ such that it [[intersection|intersects]] only finitely many elements of the cover, hence such that  $U_x \cap U_i \neq \emptyset$ for only a [[finite number]] of $i \in I$.

=--

+-- {: .num_defn #RefinementOfOpenCovers}
###### Definition
**([[refinement]] of [[open covers]])

Let $(X,\tau)$ be a [[topological space]], and let $\{U_i \subset X\}_{i \in I}$ be a [[open cover]].

Then a _[[refinement]]_ of this open cover is a set of open subsets $\{V_j \subset X\}_{j \in J}$ which is still an [[open cover]] in itself and such that for each $j \in J$ there exists an $i \in I$ with $V_j \subset U_i$.

=--

## Statement

\begin{theorem}
Every open cover of a metric space has a [[countably locally discrete set of subsets|$\sigma$-locally discrete]] [[refinement]].
\end{theorem}

\begin{corollary}
Every metric space $X$ has a $\sigma$-locally discrete base.
\end{corollary}

\begin{proof}
Set $\mathcal{U}_n \coloneqq \{ \text{open ball of radius }\; 1/n \;\text{ around }\;\, x \mid x \in X \}$. For each $n$ let $\mathcal{V}_n$ be a $\sigma$-locally discrete refinement of $\mathcal{U}_n$. By a diagonal argument the family $\mathcal{V} \coloneqq \bigcup_n \mathcal{V}_n$ is also $\sigma$-locally discrete. Moreover $\mathcal{V}$ is a base since for each point $x$ the balls of radius $1/n$ form a [[neighborhood base]].
\end{proof}

## Related statements

* [[metrization theorem]]

* [[metric spaces are paracompact]]

* [[paracompact Hausdorff spaces are normal]]

* [[locally compact and sigma-compact spaces are paracompact]]

* [[second-countable regular spaces are paracompact]]

* [[regular Lindelöf spaces are paracompact]]

* [[CW-complexes are paracompact Hausdorff spaces]]

* [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]]

## References

* [[Ryszard Engelking]], _General Topology_, Heldermann Verlag Berlin, 1989.

* K. Kuratowski, _Topology_, 2014, vol. 1.


open geometric morphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[topology|point set topology]], an [[open map]] is a [[continuous map]] that sends [[open sets]] to open sets. The notion of an **open geometric morphism** is a generalization of this notion from topology to [[topos theory]].

From a logical perspective, a [[geometric morphism]] $f:\mathcal{F}\to\mathcal{E}$ is open iff it preserves the interpretation of [[first order logic]]. This contrasts with general geometric morphisms which are only bound to preserve [[geometric logic]].

## Definition

A [[geometric morphism]]

$$
  f := (f^* \dashv f_*) : \mathcal{F} \to \mathcal{E}
$$

is called _open_ if the following equivalent conditions hold

* the [[localic reflection]] of $f$ over $\mathcal{E}$ is an [[overt space | overt]] internal locale in $\mathcal{E}$;

* the [[inverse image]] $f^*$ is a [[Heyting functor]], hence preserves [[first order logic]].

## Examples

* [[locally connected geometric morphisms]] are open. (cf. [Johnstone (2002)](#J02), p.650)

## Properties

+-- {: .num_prop}
###### Proposition
A [[geometric morphism]] $f:\mathcal{F}\to\mathcal{E}$ is open iff the canonical map $\lambda:\Omega_\mathcal{E}\to f_\ast(\Omega_\mathcal{F})$ of poset objects in $\mathcal{E}$ has an internal left adjoint $\mu :f_\ast(\Omega_\mathcal{F})\to\Omega_\mathcal{E}$.
=--

(cf. [Mac Lane-Moerdijk (1994)](#MM94), p.502)

+-- {: .num_prop}
###### Proposition
A [[geometric morphism]] $f:\mathcal{F}\to\mathcal{E}$ is open iff the pullback of any [[bounded geometric morphism]] with codomain $\mathcal{E}$ is [[skeletal geometric morphism|skeletal]] iff the pullback of any [[localic geometric morphism]] with codomain $\mathcal{E}$ is [[skeletal geometric morphism|skeletal]].
=--

This result appears as corollary 4.9 in [Johnstone (2006)](#J06).

## Related entries

* [[open morphism]]
* [[open subtopos]]
* [[proper geometric morphism]]

## References

* [[Peter Johnstone]], _Open maps of toposes_, Manuscripta Math. **31** no.1-3 (1980) pp.217-247. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/pdf/?PPN=GDZPPN002222744), [doi](https://doi.org/10.1007/BF01303275))

* {#J02}[[Peter Johnstone]], _[[Sketches of an Elephant]] vol.II_ , Oxford UP 2002. (section C3.1, pp.606-625)

* {#J06}[[Peter Johnstone]], _Complemented sublocales and open maps_ , Annals of Pure and Applied Logic **137** (2006) pp.240–255.

* [[André Joyal]], [[Myles Tierney]], _An extension of the Galois theory of Grothendieck_, Mem. Amer. Math. Soc. **309**
(1984).

* {#MM94}[[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994². (sections IX.6-8, pp.493ff; X.3, pp.535-538)

[[!redirects open geometric morphisms]]


open immersion of schemes

#Contents#
* table of contents
{:toc}

## Definition

A morphism $f:X\to Y$ of [[schemes]] is an __open immersion__ if the underlying morphism of topological spaces is a [[homeomorphism]] onto an open [[image]] and the [[comorphism]] $f^\sharp : \mathcal{O}_Y\to f_*\mathcal{O}_X$ is an [[isomorphism]] of sheaves when restricted to the image of $f$. In other words, an open immersion is a morphism of schemes which decomposes uniquely into an isomorphism of schemes and the identity [[inclusion morphism|inclusion]] of an [[open subscheme]].

## Properties

+-- {: .num_prop}
###### Proposition

Every open immersion of schemes is an [[étale morphism of schemes]]

=--

(e.g. [[The Stacks Project|Stacks Project, lemma 28.37.9]])

## Examples

+-- {: .num_example}
###### Example

For $R$ a [[ring]], $S \hookrightarrow U(R)$ a [[multiplicative subset]],  and $R \longrightarrow R[S^{-1}]$ the projection onto the [[localization of a commutative ring|localization]] at $S$, then the [[Isbell duality|formal dual]] map on [[spectrum of a commutative ring|spectra]] $Spec(R[S^{-1}]) \longrightarrow Spec(R)$ is an open immersion.

These are the _standard opens_ that define the [[Zariski topology]] on [[algebraic varieties]]

=--

## Related concepts

* [[closed immersion of schemes]]

* [[immersion]], [[open map]]

* [[closed immersion of schemes]]

* [[Zariski topology]]

[[!redirects open immersion]]
[[!redirects open immersions]]

[[!redirects open immersions of schemes]]

category: algebraic geometry


Open Logic Project


open manifold

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[manifold]] is called *open* if

1. it has no [[boundary]],

1. all its [[connected components]] are non-[[compact topological space|compact]].

In contrast, a manifold without boundary which is compact is called a *[[closed manifold]]*.

## Related concepts

* [[microflexible sheaf]]

## Literature

See also

* Wikipedia, *[Closed manifold](https://en.wikipedia.org/wiki/Closed_manifold)*

[[!redirects open manifolds]]


open map

> This page is about the concept in [[topology]]. For the more general concept see at _[[open morphism]]_.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #OpenMap}
###### Definition
**([[open maps]] and [[closed maps]])**

A [[continuous function]] $f \colon (X,\tau_X) \to (Y, \tau_Y)$ between [[topological spaces]]  is called

*  an _[[open map]]_ if the [[image]] under $f$ of an [[open subset]] of $X$ is an open subset of $Y$;

* a _[[closed map]]_ if the [[image]] under $f$ of a [[closed subset]] of $X$  is a closed subset of $Y$.

=--

## Examples

+-- {: .num_example #OpenClosedImageProjection}
###### Example
**([[image]] projections of [[open map|open]]/[[closed maps]] are themselves open/closed)**

If a [[continuous function]] $f \colon (X,\tau_X) \to (Y,\tau_Y)$ is an [[open map]] or [[closed map]] (def. \ref{OpenMap})
then so is its [[image]] projection $X \to f(X) \subset Y$, respectively, for $f(X) \subset Y$ regarded with its
[[subspace topology]].

=--

+-- {: .proof}
###### Proof

If $f$ is an open map, and $O \subset X$ is an open subset, so that $f(O) \subset Y$ is also open in $Y$, then, since $f(O) = f(O) \cap f(X)$, it is also still open in the subspace topology, hence $X \to f(X)$ is an open map.

If $f$ is a closed map, and $C \subset X$ is a closed subset so that also $f(C) \subset Y$ is a closed subset, then the [[complement]] $Y \backslash f(C)$ is open in $Y$ and  hence $(Y \backslash f(C)) \cap f(X) = f(X) \backslash f(C)$ is open in the subspace topology, which means that $f(C)$ is closed in the subspace topology.

=--

+-- {: .num_example #ProjectionsAreOpenContinuousFunctions}
###### Example
**([[projections]] out of [[product spaces]] are open maps)**

For $(X_1,\tau_{X_1})$ and $(X_2,\tau_{X_2})$ two [[topological spaces]], then the projection maps

$$
  pr_i \;\colon\; (X_1 \times X_2, \tau_{X_1 \times X_2}) \longrightarrow (X_i, \tau_{X_i})
$$

out of their [[product topological space]]

$$
  \array{
    X_1 \times X_2 &\overset{pr_1}{\longrightarrow}& X_1
    \\
    (x_1, x_2) &\overset{\phantom{AAA}}{\mapsto}& x_1
  }
$$

$$
  \array{
    X_1 \times X_2 &\overset{pr_2}{\longrightarrow}& X_2
    \\
    (x_1, x_2) &\overset{\phantom{AAA}}{\mapsto}& x_2
  }
$$

are [[open maps|open]] [[continuous functions]]  (def. \ref{OpenMap}).

This is because, by definition, every open subset $O \subset X_1 \times X_2$ in the product space topology is a union of products of
open subsets $U_i \in X_1$ and $V_i \in X_2$ in the factor spaces

$$
  O = \underset{i \in I}{\cup} \left( U_i \times V_i  \right)
$$

and because taking the image of a function preserves unions of subsets

$$
  \begin{aligned}
    pr_1\left(
      \underset{i \in I}{\cup} \left( U_i \times V_i \right)
    \right)
    & =
    \underset{i \in I}{\cup}
    pr_1
    \left(
      U_i \times V_i
    \right)
    \\
    & =
    \underset{i \in I}{\cup} U_i
  \end{aligned}
  \,.
$$

=--

\begin{proposition}\label{LocalHomeosAreOpenMaps}
  A [[local homeomorphism]] is an [[open map]].
\end{proposition}
\begin{proof}
  Let $f \colon X \to Y$ be a [[local homeomorphism]] and $U \subset X$ an [[open subset]]. We need to see that the [[image]] $f(U) \subset Y$ is an open subset of $Y$.
For this we may equivalently show that each $y \in f(U)$ has an [[open neighbourhood]] inside $f(U)$.

But since any function is [[surjection|surjective]] onto its [[image]], there exists $x \in U$ with $f(x) = y$.
By local homeomorphy of $f$, this $x \in X$ has an  [[open neighbourhood]] $U_x \subset X$ with $f_{|U_x} \colon U_x \to f(U_x)$ a [[homeomorphism]]. Since $U \cap U_x \subset U_x$ is an [[open neighbourhood]] of $x$ in $U_x$, the homeomorphy of $f_{|U_x}$ implies that $f(U \cap U_x) \subset f(U)$ is an open neighbourhood of $f(x) = y$.
\end{proof}

## Properties
 {#Properties}

\begin{proposition}\label{PreimagesOfOpenMapsPreserveInteriors}
**([[preimages]] of [[open maps]] preserve [[topological interiors]])**
\linebreak
For $f \,\colon\, X \longrightarrow Y$  an [[open map]] and $U \subset X$ a [[subset]], the [[preimage]] under $f$ of the [[interior]] of $U$ is the [[interior]] of the preimage of all of $U$:
$$
  \big( f^{-1}(U) \big)^\circ
  \;=\;
  f^{-1}\big( U^\circ\big)
  \,.
$$
\end{proposition}
\begin{proof}
In one direction, the inclusion
$$
  f^{-1}\big( U^\circ\big)
  \;\subset\;
  \big( f^{-1}(U) \big)^\circ
$$
follows since the left hand side is an [[open subset]] of $f^{-1}(U) \subset X$ by [[continuous map|continuity]] of $f$, while the right hand side is the largest such subset, by definition.

In the other direction, the inclusion
$$
  \big( f^{-1}(U) \big)^\circ
  \;\subset\;
  f^{-1}\big( U^\circ\big)
$$
is equivalent (see [there](interactions+of+images+and+pre-images+with+unions+and+intersections#PreimageRightAdjointToImage)) to the inclusion
$$
  f\Big( \big( f^{-1}(U) \big)^\circ \Big)
  \;\subset\;
  U^\circ
  \,.
$$
This follows since now the left hand side is an open subset of $f(U)$ by [[open map|open-ness]] of $f$, while the right hand side is again the largest open subset of $U$, by definition.
\end{proof}

And dually:
\begin{proposition}\label{PreimagesOfOpenMapsPreserveClosures}
**([[preimages]] of [[open maps]] preserve [[topological closure]])**
\linebreak
If $f \,\colon\, X \longrightarrow Y$ is an [[open map]], and $U \subset X$ a [[subset]] with [[topological closure]] $\overline{U}$, then the [[preimage]] $f^{-1}\big( \overline{U}\big)$ of the [[topological closure]] is the [[topological closure]] of the [[preimage]] of $U$:
$$
  \overline{f^{-1}(U)}
  \;=\;
  f^{-1}\big(  \overline{U} \big)
  \,.
$$
\end{proposition}
\begin{proof}
Noticing that

1. the topological closure is equivalently the [[complement]] of the [[topological interior]] of the [[complement]] (see [there](closed+subspace#RelationToInteriorSubspaces)):

   $$
      \overline{U}
      \,=\,
      X \setminus (X \setminus U)^\circ
   $$

1. [[preimages]] evidently preserve [[complements]]

it is sufficient to observe that forming preimages of open maps preserves interiors. This is the statement of Prop. \ref{PreimagesOfOpenMapsPreserveInteriors}.
\end{proof}

## Related concepts

* [[open morphism]]

* [[embedding of topological spaces]]

* [[closed map]], [[proper map]]

[[!redirects open maps]]
[[!redirects open mapping]]
[[!redirects open mappings]]