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
Let Top be a category of topological spaces and $B$ an object in $Top$ (the ‘base’ space). The slice category $Top/B$ is called the category of (topological) spaces over $B$ (or sometimes simply bundles).
An étalé space (or étale map) over $B$ is an object $p:E\to B$ in $Top/B$ such that $p$ is a local homeomorphism: that is, for every $e\in E$, there is an open set $U \ni e$ such that the image $p(U)$ is open in $B$ and the restriction of $p$ to $U$ is a homeomorphism $p|_U: U \to p(U)$.
The set $E_x = p^{-1}(x)$ where $x\in B$ is called the stalk of $p$ over $x$.
The underlying set of the total space $E$ (from ‘espace’) is the union of its stalks (notice that we do not say fiber!). $p$ is sometimes refered to as the projection.
Let $p:E\to B$ be in $Top/B$. The (local) sections of $p$ over an open set $U\subseteq B$ are the continuous maps $s:U\to E$ such that $p\circ s = \mathrm{id}_U$. It is an elementary but central fact that for an étale map $p$, the images of local sections form a base for the topology of the total space $E$. The topology of $E$ is then typically non-Hausdorff.
The set of sections of $p$ over $U$ is denoted by $\Gamma_U p = (\Gamma p)(U) = \Gamma_U E = (\Gamma E)(U)$ and may be shown to extend to a functor $\Gamma : Top/B\to PShv_B$ where $PShv_B$ is the category of presheaves over $B$. The functor $\Gamma$ has a left adjoint $L : PShv_B\to Top/B$, whose essential image is the full subcategory $Et/B$ of étalé spaces over $B$. The essential image of the functor $\Gamma$ is the category of sheaves $Shv_B$ over $B$, and this adjunction restricts to an equivalence of categories between $Et/B$ and $Shv_B$ (that is, it is an idempotent adjunction).
If $P:Open(X)^{op}\to Set$ is a sheaf, then one sometimes calls the total space $E(P)$ of the étalé space $L(P) = (E(P)\to B)$ the space of the sheaf $P$, having in mind the adjoint equivalence above. (This is also called the sheaf space or the display space; compare also a display morphism of contexts.) The associated sheaf functor $a:PShv_B\to Shv_B\hookrightarrow PShv_B$ decomposes as $a = \Gamma\circ L$, and $a$ may be considered as an endofunctor part of an idempotent monad in $PShv_B$ whose corresponding reflective subcategory is $Shv_B$.
(e.g. MacLane-Moerdijk, section II.5, II.6)
Every covering space (even in the more general sense not requiring any connectedness axiom) is étalé but not vice versa:
for a covering space the inverse image of some open subset in the base $B$ needs to be, by the definition, a disjoint union of homeomorphic open sets in $E$; however the ‘size’ of the open neighborhoods over various $e$ in the same stalk required in the definition of étalé space may differ, hence the intersection of their projections does not need to be an open set, if there are infinitely many points in the stalk.
even if the the stalks of the étalé space are finite, it need not be locally trivial. For instance the disjoint union $\coprod_i U_i$ of a collecton of open subsets of a topological space $X$ with the obvious projection $(\coprod_i U_i) \to X$ is étale, but does not have a typical fiber: the fiber over a given point has cardinality the number of open sets $U_i$ that contain this particular point.
In French, the verb ‘étaler’ means, roughly, to spread out; ‘-er’ becomes ‘-é’ to make a past participle. So an ‘espace étalé’ is a space that has been spread out over $B$. On the other hand, ‘étale’ is a (relatively obscure, distantly related) nautical adjective that can be translated as ‘calm’ or ‘slack’. So a ‘fonction étale’ is a slack function, one which is kind of a homeomorphism, but perhaps only locally.
To quote from the Wiktionnaire française:
‘étale’ qualifie la mer qui ne monte ni ne descend à la fin du flot ou du jusant (‘flot’ = ‘flow’ and ‘jusant’ = ‘ebb’).
There is an interesting stanza from a song of Léo Ferré:
Et que les globules figurent
Une mathématique bleue,
Sur cette mer jamais étale
D’où me remonte peu à peu
Cette mémoire des étoiles.
— (Léo Ferré, La mémoire et la mer)
He also mentions geometry and ‘théorème’ elsewhere in the song.