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