# Gracious Living

Countability Axioms
December 31, 2010, 01:55
Filed under: Math, Topology | Tags: , , , ,

So far we’ve seen two basic families of properties of topological spaces.  The connectedness axioms tell us in what ways it is possible to break our space apart into pieces.  The compactness axioms tell us how bounded the space is.  What we’re going to look at today is a set of axioms that deal with cardinality.  It should be mentioned that topology, for the most part, doesn’t really care about large cardinals — at most, we’re dealing with $\aleph_1$, the cardinality of our favorite counterexample $\omega_1$, and $c$, the cardinality of the reals.  These are equal if we accept the continuum hypothesis, and in either case we often talk about them in terms of countable subsets — sequences and the like.  The reason that countability is so important is that the properties we’re about to study are typical of metric spaces, and metrizability is a central question of point-set topology.

In the interests of total confusion, let’s discuss metrics, with have almost nothing to do with measures, despite the similarity in name!  Measures have to do with the sizes of sets of points, and are defined only on $\sigma$-algebras.  Metrics have to do with the distance between points, and are everywhere defined.  Metric topology was perhaps the earliest field of topology to be studied, and so it’s not surprising that a metric on a space will give you a natural topology.  Going the other way, from topologies to metrics, was a central problem of point-set topology in the 20th century.