From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: compactness of metric spaces  WAS:cardinality
Date: 2 Nov 1994 19:34:01 GMT

In article <1994Nov2.035053.18057@princeton.edu>,
Terry Tao <tao@math.Princeton.EDU> wrote:
>(Hmm.  I wonder if continuous bijections preserve completeness...)
Take the identity map from the open interval (0,1) to itself, using
the discrete metric in the domain and the usual metric in the codomain.
Any map out of a discrete metric space is continuous.
Any discrete metric space is complete (no non-constant Cauchy sequences)
Identity maps are always bijections.
(0,1) is not complete.

Lesson 1: when dealing with continuous maps, just think about factoring
them in three steps: X -> X/~ (a continuous surjection), where
x1 ~ x2 if f(x1)=f(x2);  then  X/~ -> f(X) (a continuous bijection);
then f(X) -> Y (inclusion of a subspace, a continuous injection). The
middle map can be viewed as the identity map on a set but with a
topology in the domain which is possibly finer than the topology on
the codomain. So "every" question about continuous maps boils down to
a study of quotient maps, inclusions, and (the one most forgotten) the
identity maps to a weakened topology.

Lesson 2: You know how compactness behaves already. For metric spaces,
compactness is equivalent to, among other characterizations,
completeness + total boundedness (for any  r > 0 , X can be covered by
a finite number of balls of radius  r). An infinite discrete space is
bounded but not totally bounded. Nice reference: Singer+Thorpe,
Lecture notes in elementary geometry and topology, Springer UTM.

dave