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
Let $X$ be a topological space which is
locally compact (in the sense that every point has an open neighbourhood whose topological closure is compact),
Then $X$ is sigma-compact.
In particular then $X$ is also paracompact since locally compact and sigma-compact spaces are paracompact.
We need to produce a countable cover of $X$ by compact subspaces.
By second-countability there exists a countable base of open subsets
By local compactness, every point $x \in X$ has an open neighbourhood $V_x$ whose topological closure $Cl(V_x)$ is compact.
By definition of base of a topology, there exists $B_x \in \beta$ such that ${x} \subset B_x \subset V_x$, hence $Cl(B_x) \subset Cl(V_x)$. Since $Cl(V_x)$ is compact by assumption, and since closed subspaces of compact spaces are compact it follows that $B_x$ is compact.
Applying this for each point yields that
But since there is only a countable set of base elements $B$ to begin with, there is a countable subset $J \subset X$ such that
Hence
is a countable cover of $X$ by compact subspaces.
Last revised on September 1, 2023 at 16:43:09. See the history of this page for a list of all contributions to it.