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 topological space $(X,\tau)$ is called a Kolmogorov space if it satisfies the $T_0$-separation axiom, hence if for $x_1 \neq x_2 \in X$ any two distinct points, then at least one of them has an open neighbourhood $U_{x_i} \in \tau$ which does not contain the other point.
the main separation axioms
number | name | statement | reformulation |
---|---|---|---|
$T_0$ | Kolmogorov | given two distinct points, at least one of them has an open neighbourhood not containing the other point | every irreducible closed subset is the closure of at most one point |
$T_1$ | given two distinct points, both have an open neighbourhood not containing the other point | all points are closed | |
$T_2$ | Hausdorff | given two distinct points, they have disjoint open neighbourhoods | the diagonal is a closed map |
$T_{\gt 2}$ | $T_1$ and… | all points are closed and… | |
$T_3$ | regular Hausdorff | …given a point and a closed subset not containing it, they have disjoint open neighbourhoods | …every neighbourhood of a point contains the closure of an open neighbourhood |
$T_4$ | normal Hausdorff | …given two disjoint closed subsets, they have disjoint open neighbourhoods | …every neighbourhood of a closed set also contains the closure of an open neighbourhood … every pair of disjoint closed subsets is separated by an Urysohn function |
A space $X$ is $T_0$ precisely when the unit of the continuation monad for the Sierpinski space $\eta : X \to \Sigma^(\Sigma^X)$ is a monomorphism.
The separation conditions $T_0$ to $T_4$ may equivalently be understood as lifting properties against certain maps of finite topological spaces, among others.
This is discussed at separation axioms in terms of lifting properties, to which we refer for further details. Here we just briefly indicate the corresponding lifting diagrams.
In the following diagrams, the relevant finite topological spaces are indicated explicitly by illustration of their underlying point set and their open subsets:
points (elements) are denoted by $\bullet$ with subscripts indicating where the points map to;
boxes are put around open subsets,
an arrow $\bullet_u \to \bullet_c$ means that $\bullet_c$ is in the topological closure of $\bullet_u$.
In the lifting diagrams for $T_2-T_4$ below, an arrow out of the given topological space $X$ is a map that determines (classifies) a decomposition of $X$ into a union of subsets with properties indicated by the picture of the finite space.
Notice that the diagrams for $T_2$-$T_4$ below do not in themselves imply $T_1$.
(Lifting property encoding $T_0$)
The following lifting property in Top equivalently encodes the separation axiom $T_0$:
(Lifting property encoding $T_1$)
The following lifting property in Top equivalently encodes the separation axiom $T_1$:
(Lifting property encoding $T_2$)
The following lifting property in Top equivalently encodes the separation axiom $T_2$:
(Lifting property encoding $T_3$)
The following lifting property in Top equivalently encodes the separation axiom $T_3$:
(Lifting property encoding $T_4$)
The following lifting property in Top equivalently encodes the separation axiom $T_4$:
(Kolmogorov quotient)
Let $(X,\tau)$ be a topological space. Consider the relation on the underlying set by which $x_1 \sim x_1$ precisely if neighther $x_i$ has an open neighbourhood not containing the other. This is an equivalence relation. The quotient topological space $X \to X/\sim$ by this equivalence relation is a $T_0$-space.
This construction is the reflector exhibiting Kolmogorov spaces as a reflective subcategory of the category Top of all topological spaces.
Last revised on August 12, 2022 at 15:57:22. See the history of this page for a list of all contributions to it.