CHAPTER 5 SEPARATION AXIOMS
We have observed instances of topological statements which, although true for all metric (and metrizable) spaces, fail for some other topological spaces. Frequently, the cause of failure can be traced to there being `not enough open sets' (in senses to be made precise). For instance, in any metric space, compact subsets are always closed; but not in every topological space, for the proof ultimately depends on the observation
`givenwhich is true in a metric space (e.g. put, it is possible to find disjoint open sets G and H with
and
'
, 
where 
) but fails in, for example,
a trivial space 
.
What we do now is to see how `demanding certain minimum levels-of-supply of open sets' gradually eliminates the more pathological topologies, leaving us with those which behave like metric spaces to a greater or lesser extent.
Spaces
(Hausdorff) Spaces
Spaces
Spaces
Spaces