From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: definition of topological space
Keywords: Topology; Open sets; Continuity
Organization: Computer Science Department,  Stanford University.
Date: Sat, 7 Nov 1992 04:42:33 GMT

In article <1992Nov6.091200.7105@leland.Stanford.EDU> ledwards@leland.Stanford.EDU (Laurence James Edwards) writes:
>In article <1992Nov5.165530.21866@CSD-NewsHost.Stanford.EDU>, pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>|> In article <1992Nov5.094404.15550@infodev.cam.ac.uk> rgep@emu.pmms.cam.ac.uk (Richard Pinch) writes:
>|> >Incidentally, {1,2,3} {} {1} is a perfectly good family of open sets
>|> >for a topology on {1,2,3}: but it has nothing to do with epsilons and
>|> >deltas.
>|> 
>|> Well, not nothing at all.  Any topology determines the nearness
>|> relation:  point x is NEAR set Y when x does not belong to Y but does
>|> belong to every closed set containing Y (i.e. to the CLOSURE of Y).
>|> 
>|> (The points near the INTERIOR of X (= complement of closure of
>|> complement of X) constitute the FRONTIER of X.  The BOUNDARY of X is
>|> the portion of its frontier lying within X, equivalently those points
>|> near the complement of X.  So the frontier of X is the boundary of X
>|> plus all points near X.)
>|> 
>|> Applying this to the example, 1 is a hermit (is near no set) while 2 is
>|> near {1}, {3}, and {1,3}, and 3 is near {1}, {2}, and {1,2}.  (So x
>|> being near singleton {y} need not imply that y is near singleton {x}.)
>|> 
>|> This is the topologically abstract expression of the general idea of
>|> epsilon-delta.
>
>Ok, this was one of the things that was confusing me with regard to finite
>sets. To me it seemed that determining the closure requires the definition
>of neighborhood and that neighborhood depends on the definition of near,
>but in the above near depends on the definition of closure. I'm having a
>hard time figuring out why this isn't a circular chain of definitions.
>To my untrained eye it would seem that a set in the absence of any
>ordering or distance relationships cannot be categorized as open or
>closed.
>
>Larry Edwards

No circularity, these are alternative definitions.  Let me collect
three of them here in the one place to make this clearer.  A
topological space is a set X with any one of the following equivalent
structures:

1. A set of subsets called open sets.
2. A function cl:P(X)->P(X) mapping subsets to subsets, called closure.
3. A binary relation near(x,Y) relating a point x to a subset Y.
(For a 4th, a reflexive version of 3, see Gareth McCaughan's msg today
on nbhd's.)

There are two things: axioms to go with each structure, and how to get
the other structures from any given one.

Axioms:
1. The open sets include the empty set and X, and are closed under binary
(hence finite) intersection and arbitrary union.
2. cl is monotone, idempotent, and increasing.  That is, X<Y -> cl(X)<cl(Y),
cl(cl(X))=X, X<cl(X) (where "<" denotes "subset of").
3. near(x,Y) is irreflexive, monotone in Y, and &-additive in Y.  That is,
not near(x,{x}), Y<Z and near(x,Y) and x not in Z -> near(x,Z), and 
near(x,Y&Z) iff near(x,Y)&near(x,Z) (& is intersection and conjunction).

Getting between these is probably best left as an exercise.

As far as |f(x)-f(y)| is concerned, following Mac Lane there are two
reasons to be more abstract, simplicity and unity.  The simplicity
reason is that when an argument goes through with a subset of the
assumptions, it only clutters things up to have the other assumptions
built into your proof.  The unity reason is that that argument is also
applicable to other models of your minimal assumptions, in which case
insisting on keeping the redundant assumptions in your proof needlessly
rules out these other applications.  I can think of no better
definition of mathematics than as the subject of simplicity and unity
of thought.

The way I think intuitively about arbitrary topologies (i.e. not even
assuming T0) is to visualize not just the open sets by the themselves
but always paired with their complementary closed sets.  That way you
can think of each pair (O,C) as one of the ways the space can tear in
half, with the frontier of the tear generously given to the closed half
C.  The trivial tear (one side empty) is *always* legal.

From this perspective it can be seen that a topological space could
just as well be defined as the pairs (Y,Z) such that Y is *not* open
(and hence Z is not closed).  That is, give all the ways in which the
space is "stuck together".  (So the discrete topology, all sets open,
says that the space "crumbles into dust" in the sense that it is not
held together in any way, while the coarse topology, only 0 and X open,
says it is titanium and there is no way it can come apart except
trivially.  The usual topology on the real plane says you can tear it
apart in relatively tame ways, but ripping out the rationals is
forbidden, the plane is a bit too sticky for that.)

This view has the nice feature that you can define a continuous map
f:X->Y to be one that preserves stickiness.  If (U,V) is one of the
ways space X cannot fall apart, then Y cannot fall apart with one half
containing f(U) and the other half containing f(V).  This neatly
captures the idea that continuous maps can't do their own tearing, they
can only tear where the domain of f already permits it.

The meaning of the T0 axiom is that there are no identical twins.
Identical twins are two points having the same membership status in
every open set: both in or both out.  T0 does for topological spaces
what antisymmetry does for partially ordered sets, namely it prevents
equivalence classes having more than one element.  (There is a one-one
correspondence between finite topologies and finite preorders
(reflexive transitive binary relations), whence a finite T0 topology is
exactly a finite partial order.  In the infinite case one talks of an
ordered topological space.  The partial order defined by cl is x<=y
just when cl({x}) subset cl({y}).)

The stronger T1 axiom says that the only good partial order is a
discrete partial order: the closure of singleton cannot lie inside the
closure of a different singleton, so x <= y (as defined above) iff
x=y.  (This is equivalent to saying that all singletons are closed.)
Since 99.9% of topology is done with T1 spaces this is why you don't
hear about ordered spaces, nontrivial ones don't arise in practice.
(This is not universally true however: the Stone duality of
distributive lattices and Stone spaces requires the full generality of
T0 spaces, and many applications of topology in computer science
similarly violate T1.)

So finite T1 topologies are discrete (since T0 topologies are partial
orders and we disallow nontrivial x <= y).  The coarsest possible
infinite T1 topology has for its closed sets exactly the finite sets
(and of course the whole space).  Hence the open sets are the cofinite
sets (finite complements), and hence no two open sets are disjoint.  If
we think of open sets as neighborhoods this means that any two
neighborhoods of two distinct points must intersect.  But real geometry
isn't like that, any two points on the plane have disjoint
neighborhoods, that is, can be housed off from each other, so we call
this the Hausdorff or T2 property.

Locales are nice too, but I think I'll stop while I'm ahead (if I am).
-- 
Vaughan Pratt				There's no truth in logic, son.