topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
A filter space is a generalisation of a topological space based on the concept of convergence of filters (or nets) as fundamental. The category of filter spaces is a quasitopos and may be thought of as a nice category of spaces that includes Top as a full subcategory. Filter spaces include convergence spaces, Choquet spaces, and Kuratowski limit spaces as full sub-quasitoposes.
A filter space is a set $S$ together with a relation $\to$ from $\mathcal{F}S$ to $S$, where $\mathcal{F}S$ is the set of filters on $S$; if $F \to x$, we say that $F$ converges to $x$ or that $x$ is a limit of $F$. This must satisfy some axioms:
The definition can also be phrased in terms of nets; a net $\nu$ converges to $x$ if and only if its eventuality filter converges to $x$.
The morphisms of filter spaces are the continuous functions; a function $f$ between filter spaces is continuous if $F \to x$ implies that $f(F) \to f(x)$, where $f(F)$ is the filter generated by the filterbase $\{F(A) \;|\; A \in F\}$. In this way, filter spaces form a concrete category $Filt$, which is a quasitopos.
A filter space that satisfies an additional directedness criterion is precisely a convergence space; see there for a variety of intermediate notions leading up to ordinary topological spaces.