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 compact subspace $K$ of a Hausdorff topological space $X$ is a closed subspace.
Let $x \in X \backslash K$ be any point of $X$ not contained in $K$. We need to show that there exists an open neighbourhood of $x$ in $X$ which does not intersect $K$.
By assumption that $X$ is Hausdorff, there exist for each $y \in K$ disjoint open neighbourhoods $y \subset U_y \subset X$ and $x \subset V_y \subset X$. Clearly the union of all the $U_y$ is an open cover of $K$
Hence by assumption that $K$ is compact, there exists a finite subset $S \subset K$ of points in $K$ such that the $U_s$ for $s \in S$ still cover $K$:
Since $S$ is finite, the intersection
is still open, and by construction it is disjoint from all $U_y$ for $y \in S$, hence in particular disjoint from $K$, and it contains $x$. Hence $U_x$ is an open neighbourhood of $x$ as required.
The statement of prop. does not hold with the Hausdorff condition replaced by weaker separation assumptions.
To see this, consider a dense subspace of a topological space. For instance the affine scheme Spec(Z) is $T_1$ but the conclusion of the proposition does not hold for the generic point.
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
maps from compact spaces to Hausdorff spaces are closed and proper
Last revised on April 16, 2017 at 07:42:46. See the history of this page for a list of all contributions to it.