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
The specialisation order is a way of turning any topological space $X$ into a preordered set (with the same underlying set).
Given a topological space $X$ with topology $\mathcal{O}(X)$, the specialization order $\leq$ is defined by either of the following two equivalent conditions:
$x \leq y$ if and only if $x$ belongs to the topological closure of $\{y\}$; we say that $x$ is a specialisation of $y$.
$x \leq y$ if and only if $\forall_{U \colon \mathcal{O}(X)} (x \in U) \Rightarrow (y \in U).$
(Note: some authors use the opposite ordering convention.)
The specialization order on a topological space is always a preorder, because the specialization order is defined as $\forall_{U \colon \mathcal{O}(X)} (x \in U) \Rightarrow (y \in U)$. Given any two sets $A$ and $B$ and any binary relation $R(x, y)$ between $A$ and $B$, the relation $\forall_{w \colon B} R(x, w) \implies R(y, w)$ is a preorder on $A$.
$X$ is $T_0$ if and only if its specialisation order is a partial order. $X$ is $T_1$ iff its specialisation order is equality. $X$ is $R_0$ (like $T_1$ but without $T_0$) iff its specialisation order is an equivalence relation. (See separation axioms.)
Given a continuous map $f: X \to Y$ between topological spaces, it is order-preserving relative to the specialisation order. Thus, we have a faithful functor $Spec$ from the category of $\Top$ of topological spaces to the category $\ProSet$ of preordered sets.
In the other direction, to each proset $X$ we may associate a topological space whose elements are those of $X$, and whose open sets are precisely the upward-closed sets with respect to the preorder. This topology is called the specialization topology. This defines a functor
which is a full embedding; the essential image of this functor is the category of Alexandroff spaces (spaces in which an arbitrary intersection of open sets is open). Hence the category of prosets is equivalent to the category of Alexandroff spaces.
In fact, we have an adjunction $i \dashv Spec$, making $ProSet$ a coreflective subcategory of $Top$. In particular, the counit evaluated at a space $X$,
is the identity function at the level of sets, and is continuous because any open $U$ of $X$ is upward-closed with respect to $\leq$, according to the second equivalent condition of the definition of the specialization order.
This adjunction restricts to an adjoint equivalence between the categories $\Fin\Pros$ and $\Fin\Top$ of finite prosets and finite topological spaces. The unit and counit are both identity functions at the level of sets, so we in fact have an equivalence between these categories as concrete categories.
Last revised on June 6, 2023 at 08:59:05. See the history of this page for a list of all contributions to it.