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 $(X,\tau)$ be a compact topological space, and let $Y \subset X$ be a closed topological subspace. Then also $Y$ is compact.
Let $\{V_i \subset Y\}_{i \in I}$ be an open cover of $Y$. We need to show that this has a finite sub-cover.
By definition of the subspace topology, there exist open subsets $U_i$ of $X$ with
By the assumption that $Y$ is closed, the complement $X \setminus Y$ is an open subset of $X$, and therefore
is an open cover of $X$. Now by the assumption that $X$ is compact, this latter cover has a finite subcover, hence there exists a finite subset $J \subset I$ such that
is still an open cover of $X$, hence in particular intersects to a finite open cover of $Y$. But since $Y \cap ( X \setminus Y ) = \empty$, it follows that indeed
is a cover of $Y$, and in indeed a finite subcover of the original one.
Last revised on July 16, 2021 at 03:25:32. See the history of this page for a list of all contributions to it.