NetNews Usenet Archive 1992 #26/NN_1992_26.iso/spool/sci/math/14448

Text File | 1992-11-06 | 2.7 KB | 56 lines

Newsgroups: sci.math
Path: sparky!uunet!snorkelwacker.mit.edu!galois!riesz!jbaez
From: [e-mail] (John C. Baez)
Subject: Re: definition of topological space
Message-ID: <[e-mail]>
Sender: [e-mail]
Nntp-Posting-Host: riesz
Organization: MIT Department of Mathematics, Cambridge, MA
References: <[e-mail]>
Date: Thu, 5 Nov 92 21:14:26 GMT
Lines: 43

In article <[e-mail]> [e-mail] (Laurence James Edwards) writes:
>The definition of a topological space is:
>
>    :a set with a collection of subsets satisfying the conditions that
>    both the empty set and the set itself belong to the collection, the
>    union of any number of the subsets is also an element of the collection,
>    and the intersection of a finite number of the subsets is an element
>    of the collection
>
>What is the purpose of this definition? 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? 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?

Yup, that's a topological space, albeit a small and dull one.   A better
example is to take R (the real numbers) and define a set S to be open if
given x in S, all points within some small distance of x are also in S.
This is the usual topology on R.

A function f from one topological space X to another Y is defined to be
continuous if given any open set O in Y, f^{-1}O is open in X.  Here
f^{-1}O means the set of all points in X which are sent by f to points
in Y.

Now for the payoff: this definition of continuity, when applied to the
example of the real numbers given above, is equivalent to the usual
epsilon/delta definition of continuity.  That is, a function f from R to R
is continuous iff for all x and all epsilon > 0 there is a delta > 0
such that if |x-y| < delta, then |f(x) - f(y)| < epsilon.

The REAL payoff is that the definition of continuity involving abstract
topological spaces is much more general and applies to many interesting
situations in which the epsilon/delta definition does not apply.  It
even applies to rather weird topological spaces like the one you
invented.  By the way, to visualize your space, think of the points 2
and 3 as sitting right on top of each other (since no open set contains
one but not the other), and the point 1 as sitting off to the side.