nice topological space
nice category of spaces
convenient category of topological spaces
Freudenthal suspension theorem
CW-complex, Hausdorff space, second-countable space, sober space
compact space, paracompact space
connected space, locally connected space, contractible space, locally contractible space
topological vector space, Banach space, Hilbert space
point, real line, plane
sphere, ball, annulus
loop space, path space
Cantor space, Sierpinski space
long line, Warsaw circle
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A topological space is separable if it has a countable dense subset.
To be explicit, X is separable if there exists an infinite sequence a:ℕ→X such that, given any point b in X and any neighbourhood U of b, we have a i∈U for some i.
A second-countable space is separable and first-countable, but the converse need not (see Steen Seebach Example 51 ).
Many results in analysis are easiest for separable spaces. This is particularly true if one wishes to avoid using strong forms of the axiom of choice or to be predicative over the natural numbers.
first-countable topological space
second-countable topological space
separable Hilbert space
Hausdorff topological space