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
symmetric monoidal (∞,1)-category of spectra
A topological ring is a ring internal to Top, a ring object in Top:
a topological space $R$ equipped with the structure of a ring on its underlying set, such that addition and multiplication are continuous functions. Of course this makes $R$ a uniform space.
A topological field is a topological ring $K$ whose underlying ring is in fact a field and such that reciprocation $(-)^{-1}: K \setminus \{0\} \to K \setminus \{0\}$ is continuous. This latter condition is the same as demanding that the subspace topology on $K \setminus \{0\}$ induced by the embedding $K \setminus \{0\} \hookrightarrow K$ coincide with the subspace topology induced by the embedding $K \setminus \{0\} \to K \times K: x \mapsto (x, x^{-1})$. More at topological field.
In a topological ring, the closure of $\{0\}$ is an ideal. It follows that for a topological field $F$, either $0$ is a closed point (so that $F$ is $T_1$ and therefore completely regular Hausdorff, by standard arguments in the theory of uniform spaces), or is a codiscrete space.
A topological algebra over a topological ring $R$ is a topological ring $S$ together with a topological ring map $R \to S$ that makes $S$ an $R$-algebra at the underlying set level (a topological associative algebra).
The real numbers form a topological field.
Any pseudocompact ring such as the completed group ring of a profinite group is a topological ring.
For any prime $p$, the ring of p-adic integers is a topological ring.
A Banach algebra is in particular a topological algebra, hence a topological ring. Hence so is a C-star-algebra.