see also algebraic topology, functional analysis and homotopy theory
topological space (see also locale)
Kolmogorov space, Hausdorff space, regular space, normal space
connected space, locally connected space, contractible space, locally contractible space
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
compact spaces equivalently have converging subnet of every net
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
Theorems
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A quotient space is a quotient object in some category of spaces, such as Top (of topological spaces), or Loc (of locales), etc.
Quotient objects in the category $Vect$ of vector spaces also traditionally use the term ‘quotient space’, but they are really just a special case of quotient modules, very different from the other kinds of quotient space. Quotient TVSes, however, combine both aspects.
Let $X$ be a topological space and $\sim$ an equivalence relation on (the underlying set of) $X$. (Since monomorphisms in Top are just injective continuous maps, to give an equivalence relation on the underlying set of a topological space is the same as to give a congruence on that space in $Top$.) Let $Y = X/{\sim}$ be the quotient set and $q\colon X\to Y$ the quotient map.
The quotient topology, or identification topology, induced on $Y$ from $X$ says that a subset $U\subseteq Y$ is open if and only if the preimage $q^{-1}(U)\subseteq X$ is open. With this topology $Y$ is a quotient space or identification space of $X$.
Obviously, up to homeomorphism, all that matters is the surjective function $X\to Y$. For the above definition, we don’t even need it to be surjective, and we could generalize to a sink instead of a single map; in such a case one generally says final topology or strong topology. See topological concrete category.
A quotient space in $Loc$ is given by a regular subobject in Frm.
(More details needed.)
Let $f \;\colon\; X \longrightarrow Y$ be a function of sets. Then a subset $S \subset S$ is called an $f$-saturated subset (or just separated subset, if $f$ is understood) if $S$ is the pre-image of its image:
Recall that a map $q \colon X \to Y$ is open if $q(U)$ is open in $Y$ whenever $U$ is open in $X$. It is not the case that a quotient map $q \colon X \to Y$ is necessarily open. Indeed, the identification map $q \colon I \sqcup \{\ast\} \to S^1$, where the endpoints of $I$ are identified with $\ast$, takes the open point $\ast$ of the domain to a non-open point in $S^1$.
Nor is it the case that a quotient map is necessarily a closed map; the classic example is the projection map $\pi_1 \colon \mathbb{R}^2 \to \mathbb{R}$, which projects the closed locus $x y = 1$ onto a non-closed subset of $\mathbb{R}$. (This is a quotient map, by the next remark.)
It is easy to prove that a continuous open surjection $p \colon X \to Y$ is a quotient map. For instance, projection maps $\pi \colon X \times Y \to Y$ are quotient maps, provided that $X$ is inhabited. Likewise, a continuous closed surjection $p: X \to Y$ is a quotient map: $p^{-1}(U)$ is open $\Rightarrow$ $p^{-1}(\neg U)$ is closed $\Rightarrow$ $p(p^{-1}(\neg U)) = \neg U$ is closed $\Rightarrow$ $U$ is open. For example, a continuous surjection from a compact space to a Hausdorff space is a quotient map.
whose underlying function $f \colon X \longrightarrow Y$ is surjective exhibits $\tau_Y$ as the corresponding quotient topology precisely if $f$ sends open and $f$-saturated subsets in $X$ to open subsets of $Y$. By this lemma this is the case precisely if it sends closed and $f$-saturated subsets to closed subsets.