So for a writing assignment in one of my classes we are asked to discuss and prove some basic results about compact sets in general topological spaces. I like proving these things, but they dont help me understand what a compact( locally compact, paracompact,...) set in a topological space "looks like." That said I'm asking for some examples of compact (locally compact, ...) sets in a variety of topological spaces. I'm also interested in some explanation of what extra "benefits" are brought about by singling out these compact sets. For example the p-adic numbers are locally compact and locally compact things (abelian groups to be precise) are a good setting in which to carry out fourier analyis.
What does a compact set look like?
5
$\begingroup$
real-analysis
general-topology
compactness
-
6As you probably know: compact is an important notion because it singles out those spaces which share the basic properties of closed intervals of real numbers: continuous functions achieve a minimum and maximum, sequences have countable subsequences (here for simplicity of exposition I am ignoring the difference between compactness and sequential compactness), the intersection of a nested sequence of closed subsets is non-empty. Locally compact spaces are singled out because they share the property of the real numbers (or more generally of Euclidean spaces) that every point has a ... – 2010-10-20
-
2... neighbourhood which is compact, i.e. which has the basic analytic properties of closed intervals. – 2010-10-20