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
subsets are closed in a closed subspace precisely if they are closed in the ambient space
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 pseudocircle is a finite topological space which is weakly homotopy equivalent to the standard circle.
In other word, for the purposes of homotopy theory (up to weak homotopy equivalence), it is equivalent to the circle, yet it is a purely finite combinatorial creature.
The pseudocircle $\mathbb{S}$ is the topological space
whose underlying set is a $4$-set, say $\{l,r,t,b\}$ (for the โleft sideโ, โright sideโ, โtop pointโ, and โbottom pointโ of the circle)
and whose topological structure, given as the collection of open subsets, is
That is the topology generated by the base $\{\{l,r,t\}, \{l,r,b\}, \{l\}, \{r\}\}$ (which is in fact the unique minimal base and furthermore the unique minimal subbase).
As a frame, this topology is a subframe of the frame of opens of the usual circle, where the names of $l$, $r$, $t$, and $b$ are taken literally.
These also name a partition of the standard circle, and this gives a quotient map? from the circle to the pseudocircle; this map is the promised weak homotopy equivalence, see below.
The function of sets
from the standard circle to the pseudocircle, which sends
a point in the open left half of $S^1$ (thought of under the standard embedding into $\mathbb{R}^2$) to the point $l \in \mathbb{S}$;
a point in the open right half of $S^1$ to $r \in \mathbb{S}$;
the top point of $S^1$ to $t \in \mathbb{S}$;
the bottom point of $S^1$ to $b \in \mathbb{S}$
is a continuous function. Moreover, it is a weak homotopy equivalence.
See the proof of the general statement at finite topological space - properties.
The only continuous functions in the other direction, $\mathbb{S} \to S^1$ are the constant maps.