From: Brandsma
Newsgroups: sci.math
Subject: Re: Geometry vs. Topology
Date: Wed, 23 Sep 1998 12:38:23 +0200
Douglas J. Zare wrote:
> wrote:
> >What, exactly, is the distinction between 'geometry' and 'topology'? Is
> >geometry just a corner of topology like, say, the theory of metric spaces?
>
>
>
>
>
> One part of topology is point-set topology. There are researchers in this
> area, but most mathematicians see this area most frequently as lemmas
> (such as sufficient criteria for a topology to come from a metric) useful
> in other parts of topology, analysis, or even algebra. I don't know what
> is at the core of this area, and suggest that someone who studies it say
> something more about it.
>
> Douglas Zare
Again there are several branches to be considered.
In a pure form point-set topology, or general topology, has as
its objective to find relations between topological properties
(defined in terms of the open sets and set-theoretical and sometimes
algebraic or other notions), and to study how these properties behave
under certain "natural" operations, for example: what properties
are preserved by taking finite products? Are there general criteria
for such properties? Does some property imply another under certain
extra conditions?
Many other branches of mathematics do similar things for the properties
that they are interested in. Finding interesting properties is quite difficult
sometimes, and here topologists are often guided by properties that are
potentially
useful for other disciplines, e.g. things like generalisations of
metrisability,
orderability and suchlike.
One looks for instance for embedding theorems: "all spaces of type 1 can be
embedded as a subspace of type 2 in a space of type 3", compactification
theory,
and category-theretical constructions in a "topology-like" setting, etc.
Some constructions are so popular that they are branches of their own: e.g.
C_p (X) theory: with a space X one associates the space of continuous
real-valued
functions with the pointwise topology. What properties of X are implied by or
imply
what properties of C_p (X)? (linear or ring-theoretical or purely topological).
Another example is the study of Cech-Stone compactifications of spaces.
Both of these examples are useful for parts of functional analysis, e.g.
Sometimes examples and theorems only exist or do not exist under certain
set-theoretical axioms. This leads to consistency results on the general
relationships
between some properties. This is called set-theoretical topology. E.g. In some
models
hereditary separability implies hereditary Lindelo"fness for regular spaces,
and in other models of set-theory it does not.
Another important branch deals with "geometrical" properties, like dimension,
and being an AR (Absolute Retract), or locally contractible of some sort.
Much (but not all) of this is mostly relevant for metrisable spaces and
manifolds.
It has close connections with algebraic topology.
This is my (somewhat subjective) overview of "general topology", reflecting of
course my own research interests..
Hope this helped,
Henno Brandsma