From: hrubin@b.stat.purdue.edu (Herman Rubin)
Newsgroups: sci.math
Subject: Applications of general topology  Was: Re: Question: Topology of finite complements
Date: 21 Jun 1998 07:47:27 -0500

In article <6mdndv$4es$1@talia.mad.ibernet.es>,
Miguel Atencia <nospam-matencia@hch.sas.cica.es> wrote:
>Hi

>No problems, just curiosity
>I've just passed my exams on Topology and I'd like to know what general
>"abstract" topology is useful for. When I say "abstract" I mean that I
>understand that topology of metric spaces is the basis of real and complex
>analysis, but I dont know the application of topology of finite complements
>(not sure of translation)and things like that. I mean not only practical
>applications but also to other fields of math or physics.
>Comments?

This particular topology does not, to my knowledge, have much in 
the line of applications.

However, non-metric topologies are of considerable use.  The weak
and weak-* topologies on infinite-dimensional Banach spaces are
non-metrizable,  Uncountable products of spaces are non-metrizable.

Also, even when spaces are metrizable, it can happen that a metric
approach is highly confusing.  The usual topology of probability
distributions on a separable metric space is metrizable, but using
metric concepts makes it much harder to understand.  In many cases
of metric spaces, the use of the metric itself confuses the much
simpler topological concepts.

For a simple example of how non-metric topologies can be used,
lower semi-continuous functions on a space are continuous for an
appropriate T_0 topology on the real numbers, the image space.


-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558