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
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
A homeomorphism (also spelt ‘homoeomorphism’ and ‘homœomorphism’ but not ‘homomorphism’) is an isomorphism in the category Top of topological spaces.
That is, a homeomorphism $f : X \to Y$ is a continuous map of topological spaces such that there is an inverse $f^{-1}: Y \to X$ that is also a continuous map of topological spaces. Equivalently, $f$ is a bijection between the underlying sets such that both $f$ and its inverse are continuous.
For more see at Introduction to Topology -- 1 the section Homeomorphisms- 1#Homeomorphisms).
The term ‘homeomorphism’ is also applied to isomorphisms in categories that generalise $Top$, such as the category of convergence spaces and the category of locales.
Beware that a continous bijection is not necessarily a homeomorphism; that is, $Top$ is not a balanced category.
For example $\exp(2\pi i(-)) [0,1) \to S^1$ is continuous and the underlying function of sets is a bijection, but the inverse function at the level of sets is not continuous.
But see prop. 1.
(homeomorphisms are the continuous and open bijections)
Let $f \;\colon\; (X, \tau_X) \longrightarrow (Y,\tau_Y)$ be a continuous function between topological spaces. Then the following are equivalence:
$f$ is a homeomorphism;
$f$ is a bijection and a closed map.
It is clear from the definition that a homeomorphism in particular has to be a bijection. The condition that the inverse function $Y \leftarrow X \colon g$ be continuous means that the pre-image function of $g$ sends open subsets to open subsets. But by $g$ being the inverse to $f$, that pre-image function is equal to $f$, regarded as a function on subsets:
Hence $g^{-1}$ sends opens to opens precisely if $f$ does, which is the case precisely if $f$ is an open map, by definition. This shows the equivalence of the first two items. The equivalence between the first and the third follows similarly via prop. \ref{ClosedSubsetContinuity}.