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. This is equivalent to its contrapositive: for all points $x_1, x_2 \in X$, if every open neighbourhood $U_{x_i} \in \tau$ which contains one of the points also contains the other point, then the two points are equal $x_1 = x_2$.
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 |
In dependent type theory, given a type $X$, the type of subtypes of $X$ is the function type $X \to \Omega$, where $\Omega$ is the type of all propositions with the type reflector type family $P:\Omega \vdash \mathrm{El}_\Omega(P) \; \mathrm{type}$.
Given a type universe $\mathrm{Type}$, a topological space is a type $X$ with a type of subtypes $O(X)$ with canonical embedding $i_O:O(X) \hookrightarrow (X \to \Omega)$, called the open sets of $X$, which are closed under finite intersections and $\mathrm{Type}$-small unions.
Given a topological space $(X, O(X))$, we define the relation
by
By definition of the type of all propositions and its type reflector, $x \in U$ is always a h-proposition for all $x:X$ and $U:O(X)$.
The specialization order is given by the type
and for all elements $x:X$, there is an element
defined by the identity function on $x \in U$
Since $x \in U$ and $y \in U$ are both h-propositions, and h-propositions are closed under function types and dependent product types, the specialization order is also valued in h-propositions.
There is a canonical family of functions for every topological space
inductively defined on reflexivity of the identity type by
A Kolmogorov topological space or $T_0$-space is a topological space which satisfies the $T_0$-separation axiom: $\mathrm{idToSpecOrd}(x, y)$ is an equivalence of types for all $x:X$ and $y:X$.
Since the specialization order is valued in propositions, the $T_0$-separation axiom ensures that the topological space is an h-set.
Regard Sierpinski space $\Sigma$ as a frame object in the category of topological spaces (meaning: the representable functor $Top(-, \Sigma): Top^{op} \to Set$ lifts through the monadic forgetful functor $Frame \to Set$), hence as a dualizing object that induces a contravariant adjunction between frames and topological spaces. Thus for a space $X$, the frame $Top(X, \Sigma)$ is the frame of open sets; for a frame $A$, there is an accompanying topological space of points $Frame(A, \Sigma)$, a subspace of the product space $\Sigma^{{|A|}}$. The unit of the adjunction is called the double dual embedding.
A topological space $X$ is $T_0$ precisely when the double dual embedding $X \to Frame(Top(X, \Sigma), \Sigma)$ is a monomorphism.
The proof is trivial: the monomorphism condition translates to saying that for any points $x, y \in X$, if the truth values of $x \in U$ and $y \in U$ agree for every open set $U$, then $x = y$.
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 neither $x_i$ 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.
preorder | partial order | equivalence relation | equality |
---|---|---|---|
topological space | Kolmogorov topological space | symmetric topological space | accessible topological space? |
Last revised on May 25, 2023 at 14:32:05. See the history of this page for a list of all contributions to it.