see also algebraic topology, functional analysis and homotopy theory
Basic concepts
topological space (see also locale)
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
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
Basic homotopy theory
The Hawaiian earring space is a famous counterexample in algebraic topology which shows the need for care in hypotheses to develop a good theory of covering spaces.
It is an example of a space which is not semi-locally simply connected.
The Hawaiian earring space is the topological space defined to be the set
endowed with subspace topology inherited from $\mathbb{R}^2$.
More abstractly: the Hawaiian earring space can be described up to homeomorphism as the one-point compactification of a coproduct (in Top; a disjoint union space) of countably many open intervals. This shows that the specific radii converging to zero (which was taken to be $1/2^n$ above) don’t actually matter.
Every neighborhood of $(0, 0)$ has non-contractible loops inside it (this would not be the case under the CW topology, i.e., the evident quotient space of countably many circles with the coproduct or disjoint sum topology, making the result a countable bouquet of circles).
Viewed in terms of general topology, it would be hard to sell the Hawaiian earring as a genuinely “pathological space”: it is compact, Hausdorff, connected and locally path-connected, etc. It’s really through the lens of algebraic topology and particularly the classical theory of covering spaces that it seems strange (it is not a CW complex, it is not semi-locally simply connected, its fundamental group is hard to compute).