From: tycchow@athena.mit.edu (Timothy Y Chow)
Subject: Re: definition of topological space
Organization: None.  This saves me from writing a disclaimer.
Date: Thu, 5 Nov 1992 20:37:38 GMT

In article <1992Nov5.033835.5180@leland.Stanford.EDU>
ledwards@leland.Stanford.EDU (Laurence James Edwards) writes:

>The definition of a topological space is:
[definition deleted]
>What is the purpose of this definition?

To appreciate the definition you have to have had some experience with
open sets on the real line R (or more generally in R^n).  Recall that
an open ball with center O and radius r is the set of all points at a
distance *strictly* less than r from O.  So it's the set of all points
"inside" a sphere or a circle or an interval, *excluding* the boundary
points.  An open set is just an arbitrary union of open balls---not
necessarily even connected.  A closed set of the real line is typically
defined as one which contains all its limit points.

If you start working out theorems about these sets and their relationship
with functions, you'll derive facts like this:

1. The intersection of finitely many open sets is open.
2. A set is closed if and only if its complement is open.
3. A function is continuous if and only if the inverse image of every open
    set is open.  (The "inverse image" of a set U is the set of all points
    that get mapped into U.)
4. Given two closed bounded disjoint sets C_1 and C_2, there are open sets
    U_1 and U_2 such that U_1 contains C_1, U_2 contains C_2, and U_1 and
    U_2 are disjoint.

In other words, many interesting properties of R^n seem to involve only
the concepts of open and closed sets, and don't seem to have much to do
with distances or angles or vector addition and multiplication.  So what
mathematicians have done is to take the crucial properties of open and
closed sets that seem to keep coming up over and over again and write
them down as axioms.  In particular they say that a collection T of
subsets of a set X is a _topology_ if

A. The empty set is in T and X is in T.
B. The union of an arbitrary number of elements of T is again in T.
C. The intersection of finitely many elements of T is again in T.

The elements of T are called _open_sets_ and complements of open sets
are called _closed_sets_.  At this point it might be helpful to point
out that you should forget about any preconceived notions of what the
words "topology" or "open set" or "closed set" should mean.  *Anything*
that satisfies the above conditions is a topology.  (Of course, the
collection of open sets of R^n as previously defined satisfy these
conditions, so it is a topology.)

Why do we do this?  By separating out these properties of open and closed
sets we clarify our thinking.  We know that any theorems we derive from
properties A-C won't depend on any concept of distance or angle.  Also
if we happen to run across other mathematical objects satisfying the
properties A-C we will have a bunch of theorems about that space all
packaged up and ready to go---we won't have to re-prove everything in
the new context.

There is still another question: why these particular properties as
opposed to any others?  The answer is that through years of experience
mathematicians have found that the properties A-C capture pretty
accurately the basic geometrical properties of R^n that don't depend on
distances or angles.  If you have heard anything about topology (here
referring to the branch of mathematics, not a collection of sets!) you
will know that it deals with properties of objects that don't change
when you stretch or deform them (e.g., the number of holes in a donut).
Getting from this general concept to properties A-C is not at all an
obvious process, and it took mathematicians many years of trial and
error to do it.

>To the naive reader (such as myself)
>it would seem that just about any set along with one of its subsets and the
>empty set would be a topological space, e.g. it would semm to me that:
>
>{1,2,3} {} {1}
>
>is a topological space. What am I missing here?

You aren't missing anything.  This is something that satisfies the axioms,
so it is in fact a topological space.  This is always what happens when you
extract certain properties from a familiar example---you will discover that
there are many other objects that satisfy these axioms, and many of these
objects are not things you planned to invite to your party.  This is often
a good thing---these unsolicited guests often furnish counterexamples to
conjectures that you might make, and sometimes even open up new branches
of mathematics.

A classic example of this is Euclid's postulates for geometry.  For nearly
two thousand years people tried to prove his fifth axiom from his other
axioms.  Finally someone discovered an example of something that satisfied
the first four axioms but not the fifth.  The counterexample, namely the
hyperbolic plane, has resulted in a lot of interesting mathematics.

>In one math dictionary
>it is stated that this definition allows one to establish the notion of
>continuity as it applies to functions between topological spaces ...
>I don't see how. Can anyone clue me in?

Remember that in R^n we proved that a function is continuous if and only
if the inverse image of every open set is open.  Well, in topology we just
*define* a function to be continuous if the inverse image of every open set
is open.  This new definition of continuous coincides with the old one in
the case of R^n but it has the advantage of making sense in new contexts.
Using this definition we can apply theorems about continuous functions to
any other topological space that comes along.
Newsgroups: sci.math