nyuuzyou/wikis huggingface dataset:
Titanic Prime whose Digits are all 0 or 1

== Theorem ==
The integer defined as:
:$10^{641} \times \dfrac {10^{640} - 1} 9 + 1$
is a titanic prime all of whose digits are either $0$ or $1$.
That is:
:$\paren 1_{640} \paren 0_{640} 1$
where $\paren a_b$ means $b$ instances of $a$ in a string.
== Proof ==
It is clear that the digits are instances of $0$ and $1$.
It is also noted that it has $640 \times 2 + 1 = 1281$ digits, making it titanic.
== Historical Note ==
== Sources ==
* 
* : $10^{641} \times \paren {10^{640} - 1} \mathop / 9 + 1$
Category:Titanic Primes
Titanic Prime whose Digits are all 0 or 1/Historical Note

== Historical Note on Titanic Prime whose Digits are all 0 or 1 ==
According to in his of $1997$, this titanic prime was discovered by in $1984$, but this has not been corroborated.
At the time it was the largest such prime number known.
It needs to be investigated whether this record has been broken since then.
== Sources ==
* 
* : $10^{641} \times \paren {10^{640} - 1} \mathop / 9 + 1$
Category:Historical Notes
Titanic Prime whose Digits are all 9 except for one 1

== Theorem ==
The integer defined as:
:$2 \times 10^{3020} - 1$
is a titanic prime all of whose digits are $9$ except one, which is $1$.
That is:
:$1 \left({9}\right)_{3020}$
where $\left({a}\right)_b$ means $b$ instances of $a$ in a string.
== Proof ==
It is clear that the digits are instances of $9$ except for the first $1$.
It is also noted that it has $3020 + 1 = 3021$ digits, making it titanic.
== Historical Note ==
== Sources ==
* 
* : $2 \times 10^{3020} - 1$
Category:Titanic Primes
Titanic Prime whose Digits are all 9 except for one 1/Historical Note

== Historical Note on Titanic Prime whose Digits are all 9 except for one 1 ==
According to in his of $1997$, this titanic prime was discovered by in $1985$, but this has not been corroborated.
At the time it was the largest such prime number known.
It needs to be investigated whether this record has been broken since then.
== Sources ==
* 
* : $2 \times 10^{3020} - 1$
Category:Historical Notes
Titanic Prime whose Digits are all Odd

== Theorem ==
The integer defined as:
:$1358 \times 10^{3821} - 1$
is a titanic prime all of whose digits are odd.
That is:
:$1357 \paren 9_{3821}$
where $\paren a_b$ means $b$ instances of $a$ in a string.
== Proof ==
It is clear that the digits are all instances of $9$ except for the initial $1357$, all of which are odd.
It is also noted that it has $4 + 3821 = 3825$ digits, making it titanic.
== Historical Note ==
== Sources ==
* 
* : $1358 \times 10^{3821} - 1$
Category:Titanic Primes
Titanic Prime whose Digits are all Odd/Historical Note

== Historical Note on Titanic Prime whose Digits are all Odd ==
According to in his of $1997$, this titanic prime was discovered by in $1988$, but this has not been corroborated.
He also reports that it is the largest known prime whose digits are odd.
This was likely the case in $1997$, but research is needed to see if this record has since been superseded.
== Sources ==
* : $1358 \times 10^{3821} - 1$
Category:Historical Notes
Titanic Prime whose Digits are all Prime

== Theorem ==
The integer defined as:
:$7352 \times \dfrac {10^{1104} - 1} {10^4 - 1} + 1$
is a titanic prime all of whose digits are themselves prime.
That is:
:$\underbrace{7352}_{275} 7353$
== Proof ==
It is clear that the digits are instances of $7$, $3$, $5$ and $2$, and so are all prime.
It is also noted that it has $275 \times 4 + 4 = 1104$ digits, making it titanic.
== Historical Note ==
== Sources ==
* 
* 
* : $7532 \times \paren {10^{1104} - 1} / \paren {10^4 - 1} + 1$
Category:Titanic Primes
Titanic Prime whose Digits are all Prime/Historical Note

== Historical Note on Titanic Prime whose Digits are all Prime ==
According to in his of $1997$, this titanic prime was discovered by in $1988$, but this has not been corroborated.
At the time it was the largest such prime number known.
It needs to be investigated whether this record has been broken since then.
It was unfortunately misreported by as $7532 \times \dfrac {10^{1104} - 1} {10^4 - 1} + 1$, which is composite.
This mistake was propagated by , who repeated it in his of $1997$.
== Sources ==
* 
* 
* : $7532 \times \paren {10^{1104} - 1} / \paren {10^4 - 1} + 1$
Category:Historical Notes
Titanic Prime whose Digits are all Prime/Mistake

== Source Work ==
:
:The Dictionary
::$7532 \times \paren {10^{1104} - 1} / \paren {10^4 - 1} + 1$
== Mistake ==
:$7532 \times \paren {10^{1104} - 1} / \paren {10^4 - 1} + 1 \qquad \qquad$ [$1104$ digits]
== Correction ==
This was transcribed incorrectly at some point in the chain of reports.
The number given is composite, and has $6043$ as a divisor.
The correct number appears to be:
:$7352 \times \dfrac {10^{1104} - 1} {10^4 - 1} + 1$
which is indeed prime
 misreports it in a $1994$ article, and it is likely (but has not been confirmed) that he made the mistake in his $1989$ book, which contains the same factoid.
This conclusion follows from the fact that , propagating this mistake, references the book, but this has not been corroborated.
The original work by , in which this number is first reported, is still to be tracked down.
== Sources ==
* 
* 
* : $7532 \times \paren {10^{1104} - 1} / \paren {10^4 - 1} + 1$
Category:Mistakes/Curious and Interesting Numbers
Titanic Sophie Germain Prime

== Theorem ==
The integer defined as:
:$39 \, 051 \times 2^{6001} - 1$
is a titanic prime which is also a Sophie Germain prime:
== Proof ==
At $1812$ digits, it is clear by definition that this prime is titanic.
To show that it is in fact a Sophie Germain prime, we also need to check that:
:$2 \paren {39 \, 051 \times 2^{6001} - 1} + 1 = 39 \, 051 \times 2^{6002} - 1$
is also prime.
== Historical Note ==
== Sources ==
* 
* : $39,051 \times 2^{6001} - 1$
Category:Titanic Primes
Category:Sophie Germain Primes
Titanic Sophie Germain Prime/Historical Note

== Historical Note on Titanic Prime whose Digits are all 0 or 1 ==
When documented this entry in his of $1997$, this titanic prime was the largest Sophie Germain prime known.
There are now plenty of larger ones known.
== Sources ==
* 
* : $39,051 \times 2^{6001} - 1$
Category:Historical Notes
Titanic Sophie Germain Prime/Mistake 1

== Source Work ==
:
:The Dictionary
::$39,051 \times 2^{6001} - 1$
== Mistake ==
:$39,051 \times 2^{6001} - 1 \qquad \qquad$ [$1811$ digits]
== Correction ==
There are in fact $1812$ digits in this number, not $1811$.
== Sources ==
* : $39,051 \times 2^{6001} - 1$
Category:Mistakes/Curious and Interesting Numbers
Titanic Sophie Germain Prime/Mistake 2

== Source Work ==
:
:The Dictionary
::$39,051 \times 2^{6001} - 1$
== Mistake ==
:The largest known Sophie Germain prime.
This entry is misplaced in the dictionary.
It comes between:
:$10^{641} \times \paren {10^{640} - 1} \mathop / 9 + 1 \qquad \qquad$ [$1281$ digits]
and:
:$2^{4253} - 1 \qquad \qquad$ [$1281$ digits]
but itself has $1812$ digits, and so should come after $2^{4253} - 1$.
== Sources ==
* : $39,051 \times 2^{6001} - 1$
Category:Mistakes/Curious and Interesting Numbers
Titius-Bode Law

== Physical Law ==
The Titius-Bode law is the observation that the semi-major axes of successive planets of the solar system are governed by the equation:
:$a = 4 + \floor {2^{n - 1} } \times 3$
where:
:$a$ is the measure of distance such that the Earth's semi-major axes is defined as $10$
:$n = 0, 1, 2, \ldots$ for successive planets
:$\floor {2^{n - 1} }$ denotes the floor of $2^{n - 1}$.
That is, $a$ is defined by the sequence:
:$4, 7, 10, 16, 28, 52, 100, 196, \ldots$
While accurate for the planets out as far as Uranus, the law breaks down for Neptune and beyond.
== Also known as ==
The Titius-Bode Law is also seen named as Bode's Law.
== Historical Note ==
Category:Celestial Mechanics
Category:Solar System
Titius-Bode Law/Historical Note

== Historical Note on Titius-Bode Law ==
It Titius-Bode Law first appears in 's Astronomiae physicae et geometricae elementa of $1702$.
It was later picked up on by , who inserted it into his $1766$ translation of Charles Bonnet's Contemplation de la Nature.
It was noticed that, were this law to hold, there was a missing planet between Mars and Jupiter.
 followed this up initially in $1772$, in a footnote to the second edition of his Anleitung zur Kenntniss des gestirnten Himmels.
After the discovery of Uranus in $1781$, which also appeared to obey the same law, the gap between Mars and Jupiter was taken more seriously.
 proposed a systematic search for the planet which was expected to exist. In due course, Ceres was discovered in $1801$.
The observational data on Ceres was sparse, and it was soon lost when it passed into the area of the sky occupied by the sun.
However, succeeded in working out, using the method of least squares, where it was most likely to be, and when it was once more in an area of sky where it could be seen, it was found again.
The Titius-Bode Law was then widely accepted until Neptune was discovered in $1846$ and found not to satisfy it, causing the law to be discredited.
It needs to be borne in mind that the Titius-Bode Law was no more than a pattern which had been noticed, and no scientific reason had been offered for it.
In more recent times, however, it has been suggested that the gravitational influence of Jupiter and Saturn may be the source of a resonance effect of some kind.
== Sources ==
* : Chapter $\text {A}.25$: Gauss ()
Category:Historical Notes
Toeplitz Matrix/Examples/Arbitrary Order 4

== Example of Toeplitz Matrix ==
A Toeplitz matrix of order $4$ is in the form:
:$\begin {pmatrix} a & b & c & d \\ e & a & b & c \\ f & e & a & b \\ g & f & e & a \end {pmatrix}$
== Sources ==
* 
* 
Category:Examples of Toeplitz Matrices
Tonelli's Theorem/Corollary

== Corollary to Tonelli's Theorem ==
Let $\sequence {a_{n, m} }_{\tuple {n, m} \in \N^2}$ be a doubly subscripted sequence of non-negative real numbers.
Then: 
:$\ds \sum_{n \mathop = 1}^\infty \paren {\sum_{m \mathop = 1}^\infty a_{n, m} } = \sum_{m \mathop = 1}^\infty \paren {\sum_{n \mathop = 1}^\infty a_{n, m} }$
== Proof ==
Consider the measure space $\struct {\N, \map \PP \N, \mu}$ where $\mu$ is the counting measure.
From Counting Measure on Natural Numbers is Sigma-Finite, we have that $\struct {\N, \map \PP \N, \mu}$ is $\sigma$-finite.
Let $\struct {\N \times \N, \map \PP \N \otimes \map \PP \N, \mu \times \mu}$ be the product $\sigma$-algebra of $\struct {\N, \map \PP \N, \mu}$ with itself. 
Define a function $f : \N^2 \to \R$ by: 
:$\map f {n, m} = a_{n, m}$ 
for each $\tuple {n, m} \in \N^2$.
From Function Measurable with respect to Power Set, we have that $f$ is $\map \PP \N$-measurable. 
We can therefore apply Tonelli's Theorem to obtain: 
:$\ds \int \paren {\int \map f {n, m} \map {\rd \mu} m} \map {\rd \mu} n = \int \paren {\int \map f {n, m} \map {\rd \mu} n} \map {\rd \mu} m$
From Integral of Positive Function with respect to Counting Measure on Natural Numbers, this is equivalent to:
:$\ds \int \paren {\sum_{m \mathop = 1}^\infty \map f {n, m} } \map {\rd \mu} n = \int \paren {\sum_{n \mathop = 1}^\infty \map f {n, m} } \map {\rd \mu} m$
Applying Integral of Positive Function with respect to Counting Measure on Natural Numbers, we obtain: 
:$\ds \sum_{n \mathop = 1}^\infty \paren {\sum_{m \mathop = 1}^\infty \map f {n, m} } = \sum_{m \mathop = 1}^\infty \paren {\sum_{n \mathop = 1}^\infty \map f {n, m} }$
That is: 
:$\ds \sum_{n \mathop = 1}^\infty \paren {\sum_{m \mathop = 1}^\infty a_{n, m} } = \sum_{m \mathop = 1}^\infty \paren {\sum_{n \mathop = 1}^\infty a_{n, m} }$
Category:Tonelli's Theorem
Tonelli's Theorem/Lemma 1

== Lemma ==
Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be $\sigma$-finite measure spaces.
Let $\struct {X \times Y, \Sigma_X \otimes \Sigma_Y, \mu \times \nu}$ be the product measure space of $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$.
Let $E \in \Sigma_X \otimes \Sigma_Y$. 
Let: 
:$f = \chi_E$
where $\chi_E$ is the characteristic function of $E$. 
Then:
:$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_Y \paren {\int_X f^y \rd \mu} \rd \nu = \int_X \paren {\int_Y f_x \rd \nu} \rd \mu$
where: 
:$f^y$ is the $y$-horizontal section of $f$
:$f_x$ is the $x$-vertical section of $f$. 
== Proof ==
From Integral of Characteristic Function: Corollary, we then have: 
:$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \map {\paren {\mu \times \nu} } E$
From Horizontal Section of Characteristic Function is Characteristic Function of Horizontal Section, we have: 
:$f^y = \chi_{E^y}$
From Vertical Section of Characteristic Function is Characteristic Function of Vertical Section, we have: 
:$f_x = \chi_{E_x}$
We therefore have: 
and:
Then: 
and:
So, we have: 
:$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_Y \paren {\int_X f^y \rd \mu} \rd \nu = \int_X \paren {\int_Y f_x \rd \nu} \rd \mu$
Category:Tonelli's Theorem
Tonelli's Theorem/Lemma 2

== Lemma ==
Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be $\sigma$-finite measure spaces.
Let $\struct {X \times Y, \Sigma_X \otimes \Sigma_Y, \mu \times \nu}$ be the product measure space of $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$.
Let $f : X \times Y \to \overline \R$ be a positive simple function.
Then: 
:$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_Y \paren {\int_X f^y \rd \mu} \rd \nu = \int_X \paren {\int_Y f_x \rd \nu} \rd \mu$
where: 
:$f^y$ is the $y$-horizontal section of $f$
:$f_x$ is the $x$-vertical section of $f$. 
== Proof ==
Write the standard representation of $f$ as: 
:$\ds f = \sum_{k \mathop = 1}^n a_k \chi_{E_k}$
with:
:$E_1, E_2, \ldots, E_n$ pairwise disjoint $\Sigma_X \otimes \Sigma_Y$-measurable sets
:$a_1, a_2, \ldots, a_n$ non-negative real numbers.
From Horizontal Section of Simple Function is Simple Function, we have: 
:$f^y$ is a positive simple function
with: 
:$\ds f^y = \sum_{k \mathop = 1}^n a_k \chi_{\paren {E_k}^y}$
where:
:$\paren {E_1}^y, \paren {E_2}^y, \ldots, \paren {E_n}^y$ are pairwise disjoint $\Sigma_X$-measurable sets
:$a_1, a_2, \ldots, a_n$ are real numbers.
From the definition of the $\mu$-integral of a positive simple function, we have: 
:$\ds \map {I_\mu} {f^y} = \sum_{k \mathop = 1}^n a_k \map \mu {\paren {E_k}^y}$
From Integral of Positive Measurable Function Extends Integral of Positive Simple Function, we then have: 
:$\ds \int f^y \rd \mu = \sum_{k \mathop = 1}^n a_k \map \mu {\paren {E_k}^y}$
Similarly, from Vertical Section of Simple Function is Simple Function, we have: 
:$f_x$ is a positive simple function
with: 
:$\ds f_x = \sum_{k \mathop = 1}^n a_k \chi_{\paren {E_k}_x}$
where:
:$\paren {E_1}_x, \paren {E_2}_x, \ldots, \paren {E_n}_x$ are pairwise disjoint $\Sigma_X$-measurable sets
:$a_1, a_2, \ldots, a_n$ are non-negative real numbers.
From the definition of the $\nu$-integral of a positive simple function, we have: 
:$\ds \map {I_\nu} {f_x} = \sum_{k \mathop = 1}^n a_k \map \nu {\paren {E_k}_x}$
From Integral of Positive Measurable Function Extends Integral of Positive Simple Function, we then have: 
:$\ds \int f_x \rd \nu = \sum_{k \mathop = 1}^n a_k \map \nu {\paren {E_k}_x}$
We then have: 
With a view to show: 
:$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_Y \paren {\int_X f^y \rd \mu} \rd \nu$
write:
To show that: 
:$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_X \paren {\int_Y f_x \rd \nu} \rd \mu$
write: 
So:
:$\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_Y \paren {\int_X f^y \rd \mu} \rd \nu = \int_X \paren {\int_Y f_x \rd \nu} \rd \mu$
Category:Tonelli's Theorem
Top equals to Relative Pseudocomplement in Brouwerian Lattice

== Theorem ==
Let $\struct {S, \vee, \wedge, \preceq}$ be a Brouwerian lattice with greatest element $\top$.
Let $a, b \in S$.
Then
:$\top = a \to b$ $a \preceq b$
== Proof ==
=== Sufficient Condition ===
Let:
:$\top = a \to b$
By definition of reflexivity:
:$\top \preceq a \to b$
By Inequality with Meet Operation is Equivalent to Inequality with Relative Pseudocomplement in Brouwerian Lattice:
:$\top \wedge a \preceq b$
Thus
:$a \preceq b$
=== Necessary Condition ===
Let:
:$a \preceq b$
Then:
:$a \wedge \top \preceq b$
By Inequality with Meet Operation is Equivalent to Inequality with Relative Pseudocomplement in Brouwerian Lattice:
:$\top \preceq a \to b$
By definition of greatest element:
:$a \to b \preceq \top$
By definition of ordered set:
:$\top = a \to b$
== Sources ==
* 
* 
Category:Brouwerian Lattices
Top in Filter

== Theorem ==
Let $\struct {S, \preceq}$ be a bounded above ordered set.
Let $F$ be a filter on $S$.
Then $\top \in F$
where $\top$ denotes the greatest element of $S$.
== Proof ==
By definition of filter in ordered set:
:$F$ is non-empty and upper.
By definition of non-empty set:
:$\exists x: x \in F$
By definition of greatest element:
:$x \preceq \top$
Thus by definition of upper section:
:$\top \in F$
== Sources ==
* 
Category:Order Theory
Top in Ordered Set of Topology

== Theorem ==
Let $T = \left({S, \tau}\right)$ be a topological space.
Let $P = \left({\tau, \subseteq}\right)$ be an inclusion ordered set of $\tau$.
Then $P$ is bounded above and $\top_P = S$
where $\top_P$ denotes the greatest element in $P$.
== Proof ==
By definition of topological space:
:$S \in \tau$
By definition of subset:
:$\forall A \in \tau: A \subseteq S$
Hence $P$ is bounded above.
Thus by definition of the greatest element:
:$\top_P = S$
== Sources ==
* 
Category:Topology
Category:Order Theory
Top is Complete

== Theorem ==
The category of topological spaces is complete.
== Proof ==
Let $\II$ be a small category.
Let $D : \II \to \mathbf {Top}$ be a diagram in the category of topological spaces $\mathbf {Top}$.
Let $\family {\lim D, \family {\pi_i}_{i \mathop \in \II}}$ be the limit of topological spaces of $D$.
By Limit of Topological Spaces is Limit, $\family {\lim D, \family {\pi_i}_{i \mathop \in \II}}$ is a categorical limit of $D$.
Category:Topological Spaces
Category:Category Theory
Top is Meet Irreducible

== Theorem ==
Let $\left({S, \wedge, \preceq}\right)$ be a bounded above meet semilattice.
Then $\top$ is meet irreducible
where $\top$ denotes the greatest element in $S$.
== Proof ==
Let $x, y \in S$ such that
:$\top = x \wedge y$
By Meet Precedes Operands
:$\top \preceq x$ and $\top \preceq y$
By definition of greatest element:
:$x \preceq \top$
Thus by definition of antisymmetry:
:$\top = x$
== Sources ==
* 
* 
Category:Order Theory
Category:Meet Irreducible Elements
Top is Prime Element

== Theorem ==
Let $L = \struct {S, \wedge, \preceq}$ be a bounded above meet semilattice.
Then $\top$ is a prime element
where $\top$ denotes the greatest element of $L$.
== Proof ==
Let $x, y \in S$ such that
:$x \wedge y \preceq \top$
Thus by definition of greatest element:
:$x \preceq \top$ or $y \preceq \top$
== Sources ==
* 
* 
Category:Prime Elements
Top of Lattice is Unique

== Theorem ==
Let $\struct {S, \vee, \wedge, \preceq}$ be a lattice.
Then $S$ has at most one top.
== Proof ==
By definition, a top for $S$ is a greatest element.
The result follows from Greatest Element is Unique.
== Also see ==
* Bottom of Lattice is Unique
Category:Lattice Theory
Topological Classification of Uppercase Sans Serif English Letters

== Theorem ==
Recall the letters of the English language, written in a sans serif style:
They can be divided into homeomorphism classes as follows:
:$\mathsf A, \mathsf R$
:$\mathsf B$
:$\mathsf C, \mathsf G, \mathsf I, \mathsf J, \mathsf L, \mathsf M, \mathsf N, \mathsf S, \mathsf U, \mathsf V, \mathsf W, \mathsf Z$
:$\mathsf D, \mathsf O$
:$\mathsf E, \mathsf F, \mathsf T, \mathsf Y$
:$\mathsf H, \mathsf K$
:$\mathsf P$
:$\mathsf Q$
:$\mathsf X$
== Sources ==
* : $3$: Continuity generalized: topological spaces: Exercise $3.9: 23$
Category:Examples of Homeomorphisms
Topological Closure in Coarser Topology is Larger

== Theorem ==
Let $X$ be a set. 
Let $\tau_1$ and $\tau_2$ be topologies on $X$ such that:
:$\tau_1 \subseteq \tau_2$
That is, such that $\tau_1$ is coarser than $\tau_2$. 
Let $S \subseteq X$.
Then we have: 
:$\map {\cl_2} S \subseteq \map {\cl_1} S$
where $\cl_1$ and $\cl_2$ denote topological closure in $\struct {X, \tau_1}$ and $\struct {X, \tau_2}$ respectively.
== Proof ==
Let $\CC_1$ be the set of closed sets $C$ in the topological space $\struct {X, \tau_1}$ such that $S \subseteq C$.
Let $\CC_2$ be the set of closed sets $C$ in the topological space $\struct {X, \tau_2}$ such that $S \subseteq C$.
Let $C \in \CC_1$.
Then from Closed Set in Coarser Topology is Closed in Finer Topology:
:$C \in \CC_2$
Hence by definition of subset:
:$\CC_1 \subseteq \CC_2$
From Intersection is Decreasing: 
:$\bigcap \CC_2 \subseteq \bigcap \CC_1$
From the definition of topological closure:
:$\cl_1 := \bigcap \CC_1$
and:
:$\cl_2 := \bigcap \CC_2$
Hence the result:
:$\map {\cl_2} S \subseteq \map {\cl_1} S$
Category:Set Closures
Topological Closure is Closed

== Theorem ==
Let $T$ be a topological space.
Let $H \subseteq T$.
Then the closure $\map \cl H$ of $H$ is closed in $T$.
== Proof ==
From Closure of Topological Closure equals Closure:
:$\map \cl {\map \cl H} = \map \cl H$
From Set is Closed iff Equals Topological Closure, it follows that $\map \cl H$ is closed.
== Also see ==
* Topological Closure is Closure Operator
== Sources ==
* : $2.27 \, \text a$
* : $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Proposition $3.7.15 \ \text{(d)}$
* : $1$: Complex Numbers: Point Sets: $9.$
Category:Set Closures
Category:Closed Sets
Topological Closure is Closure Operator

== Theorem ==
The topological closure operator is a closure operator.
== Proof ==
=== Extensive ===
Follows from Set is Subset of its Topological Closure.
=== Increasing ===
Follows immediately from Topological Closure of Subset is Subset of Topological Closure.
=== Idempotent ===
Follows immediately from Closure of Topological Closure equals Closure.
Category:Set Closures
Category:Examples of Closure Operators
Topological Closure of Singleton is Irreducible

== Theorem ==
Let $T = \struct {S, \tau}$ be a topological space.
Let $x$ be a point of $T$.
Then:
:$\set x^-$ is irreducible
where $\set x^-$ denotes the topological closure of $\set x$.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* 
Category:Irreducible Spaces
Category:Topological Closure of Singleton is Irreducible
Topological Closure of Singleton is Irreducible/Proof 2

== Theorem ==
== Proof ==
 that
:$\set x^-$ is not irreducible.
By Set is Subset of its Topological Closure:
:$\set x \subseteq \set x^-$
By definitions of singleton and Subset:
:$x \in \set x^-$
By definition of irreducible:
:$\exists X_1, X_2 \subseteq S: X_1, X_2$ are closed
and:
By definition of union:
:$x \in X_1$ or $x \in X_2$
By definitions of singleton and subset:
:$\set x \subseteq X_1$ or $\set x \subseteq X_2$
By definition of closure :
:$\set x^- \subseteq X_1$ or $\set x^- \subseteq X_2$
By Set is Subset of Union:
:$X_1 \subseteq \set x$ and $X_2 \subseteq \set x$
Thus by definition of set equality:
:this contradicts $\set x^- \ne X_1$ and $\set x^- \ne X_2$
Category:Topological Closure of Singleton is Irreducible
Topological Closure of Subset is Subset of Topological Closure

== Theorem ==
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq K$ and $K \subseteq S$.
Then:
:$\map \cl H \subseteq \map \cl K$
where $\map \cl H$ denotes the closure of $H$.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Proposition $3.7.15 \ \text{(b)}$
Category:Subsets
Category:Set Closures
Category:Topological Closure of Subset is Subset of Topological Closure
Topological Closure of Subset is Subset of Topological Closure/Proof 1

== Theorem ==
== Proof ==
From Topological Closure is Closed, $\map \cl K$ is closed.
From Set is Subset of its Topological Closure:
:$K \subseteq \map \cl K$
By Subset Relation is Transitive, it follows that:
:$H \subseteq \map \cl K$
Hence, by definition of closure as the smallest closed set that contains $H$:
:$\map \cl H \subseteq \map \cl K$
Category:Topological Closure of Subset is Subset of Topological Closure