This entry is about the general concept. For the concept in topology see at closed map.
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
A continuous function $f \colon X \longrightarrow Y$ between topological spaces is called a closed map if the image of every closed subset in $X$ is also closed in $Y$.
Recall that $f$ is a continuous map if the preimage of every closed set in $Y$ is closed in $X$. For defining closed maps typically one restricts attention to closed continuous maps, although it also makes sense to speak of closed functions that are not continuous.
A map $f : X \to Y$ of locales is closed iff for any $u \in O(X)$ and $v \in O(Y)$ the reciprocity relation
holds. (“$\geq$” is trivially always satisfied.) The map $f_* : O(X) \to O(Y)$ is the monotone right adjoint to $f^* : O(Y) \to O(X)$, explicitly given by
A geometric morphism $f \;\colon\; \mathcal{F} \longrightarrow \mathcal{E}$ of toposes is closed iff for any object $A \in \mathcal{E}$, the induced locale homomorphism
between the spaces of subobjects of the corresponding terminal objects is closed in the sense of def. .
Let $A$ be an object of a topos $\mathcal{E}$. Then the canonical etale geometric morphism $\mathcal{E}/A \to A$ is closed iff $A$ fulfills the following condition, formulated in the internal language:
Note that this condition is satisfied for any $A$ whatsoever if the internal language of $\mathcal{E}$ is boolean.
Let $f : A \to B$ be a morphism in a topos $\mathcal{E}$. Then the induced geometric morphism $\mathcal{E}/A \to \mathcal{E}/B$ is closed iff, in the internal language of $\mathcal{E}$, the fibers of $f$ fulfill the condition displayed at the previous example.
A geometric morphism $f : \mathcal{F} \to \mathcal{E}$ is closed iff, in the internal language of $\mathcal{E}$, the unique locale map $f_* \Omega_{\mathcal{F}} \to \mathrm{pt}$ into the one-point space (given by the frame $\Omega_{\mathcal{E}}$) is closed.
Closed maps of locales and toposes are discussed in section C3.2 of
Last revised on June 11, 2017 at 20:50:39. See the history of this page for a list of all contributions to it.