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
algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
The category of quasi-topological spaces was proposed in Spanier 1963 as a substitute for the category Top of ordinary topological spaces, in order to serve as a convenient category for the purposes of algebraic topology. In particular, quasi-topological spaces form a complete and cocomplete cartesian closed category.
Today, quasi-topological spaces seem to be regarded mostly as a historical curiosity, perhaps because working topologists were never comfortable with the set-theoretic issues that accompany them. In retrospect, however, they are an impressive testament to the conceptual insight of Spanier into ideas of topos theory which were at the time (early 1960’s) barely in the air, and even not quite born yet (being an early example of quasitopos, whose name perhaps derives from Spanier’s notion, compare Dubuc & Español 2006, p. 12).
Let $\mathcal{C H}$ be the category of compact Hausdorff spaces. This may be regarded as a (large) site when equipped with the Grothendieck topology of finite open covers, in fact a concrete site.
A quasi-topological space is a (small-set valued) concrete sheaf on $\mathcal{C H}$.
The (super-large) category of quasi-topological spaces is a quasitopos (although this is not immediately obvious for size reasons — in particular, it is probably not a Grothendieck quasitopos). In particular, it is a locally cartesian closed category.
Edwin Spanier, Quasi-topologies, Duke Mathematical Journal 30, number 1 (1963) pp 1–14 (doi:10.1215/S0012-7094-63-03001-1)
Eduardo Dubuc, Luis Español, Quasitopoi over a base category (arXiv:math.CT/0612727)