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 space (such as a topological space) is second-countable if, in a certain sense, there is only a countable amount of information globally in its topology. (Change ‘globally’ to ‘locally’ to get a first-countable space.)
(second-countable topological space)
A topological space is second-countable if it has a base for its topology consisting of a countable set of subsets.
A locale is second-countable if there is a countable set $B$ of open subspaces (elements of the frame of opens) such that every open $G$ is a join of some subset of $B$. That is, we have
The weight of a space is the minimum of the cardinalities of the possible bases $B$. We are implicitly using the axiom of choice here, to suppose that this set of cardinalities (which really is a small set because bounded above by the number of open subspaces, and inhabited by this number as well) has a minimum. But without Choice, we can still consider this collection of cardinalities.
Then a second-countable space is simply one with a countable weight.
(Euclidean space is second-countable)
Let $n \in \mathbb{N}$. Consider the Euclidean space $\mathbb{R}^n$ with its Euclidean metric topology. Then $\mathbb{R}^n$ is second countable.
A countable set of base open subsets is given by the open balls $B^\circ_x(\epsilon)$ of rational radius $\epsilon \in \mathbb{Q}_{\geq 0} \subset \mathbb{R}_{\geq 0}$ and centered at points with rational coordinates: $x \in \mathbb{Q}^n \subset \mathbb{R}^n$.
A compact metric space is second-countable.
A separable metric space, e.g., a Polish space, is second-countable.
It is not true that separable first-countable spaces are second-countable; a counterexample is the real line equipped with the half-open or lower limit topology that has as basis the collection of half-open intervals $[a, b)$.
A Hausdorff locally Euclidean space is second-countable precisely it is paracompact and has a countable set of connected components. In this case it is called a topological manifold.
See at topological space this prop..
A countable coproduct (disjoint union space) of second-countable spaces is second-countable.
Countable products (product topological spaces) of second-countable spaces are second-countable.
Subspaces of second-countable spaces are second-countable.
If $X$ is second-countable and there is an open surjection $f \colon X \to Y$, then $Y$ is second-countable.
For second-countable T_3 spaces $X, Y$, if $X$ is locally compact, then the mapping space $Y^X$ with the compact-open topology is second-countable.
Cf. Urysohn metrization theorem and Polish space. I (Todd Trimble) am uncertain to what extent the $T_3$ assumption can be removed.
properties | implications |
---|---|
metrisable: topology is induced by a metric | a metric space has a $\sigma$-locally discrete base as a corollary of the fact that open covers of metric spaces have open $\sigma$-locally discrete refinements: take $\sigma$-locally discrete refinements of the covers by $1/n$-balls for $n=1,2,\ldots$. |
second-countable: there is a countable base of the topology. | Nagata-Smirnov metrization theorem |
$\sigma$-locally discrete base, i.e. $X$ has a $\sigma$-locally discrete base. | a second-countable space has a $\sigma$-locally finite base: take the the collection of singeltons of all elements of countable cover of $X$. |
$\sigma$-locally finite base, i.e. $X$ has a countably locally finite base. | second-countable spaces are separable: choose a point in each set of countable cover. |
separable: there is a countable dense subset. | second-countable spaces are Lindelöf |
Lindelöf: every open cover has a countable sub-cover. | weakly Lindelöf spaces with countably locally finite base are second countable |
weakly Lindelöf: every open cover has a countable subcollection the union of which is dense. | separable metacompact spaces are Lindelöf |
countable choice: the natural numbers is a projective object in Set. | separable spaces are weakly Lindelöf: given a countable dense subset and an open cover choose for each point of the subset an open from the cover. |
metacompact: every open cover has a point-finite open refinement. | Lindelöf spaces are trivially also weakly Lindelöf. |
first-countable: every point has a countable neighborhood base | a space with a $\sigma$-locally finite base is first countable: obviously, every point is contained in at most countably many sets of a $\sigma$-locally finite base. |
Frechet-Uryson space: the closure of a set $A$ consists precisely of all limit points of sequences in $A$ | a first-countable space is Fréchet-Urysohn: obvious |
sequential topological space: a set $A$ is closed if it contains all limit points of sequences in $A$ | obviously, a Fréchet-Uryson space is sequential |
countably tight: for each subset $A$ and each point $x\in \overline A$ there is a countable subset $D\subseteq A$ such that $x\in \overline D$. | obviously, a sequential space is countably tight. |
Last revised on April 5, 2019 at 19:51:00. See the history of this page for a list of all contributions to it.