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
(compact subspaces in Hausdorff spaces are separated by neighbourhoods from points)
Let
$(X,\tau)$ be a Hausdorff topological space;
Then for every $x \in X \backslash Y$ there exists
an open neighbourhood $U_x \supset \{x\}$;
an open neighbourhood $U_Y \supset Y$
such that
As a direct consequence compact subspaces of Hausdorff spaces are closed.
By the assumption that $(X,\tau)$ is Hausdorff, we find for every point $y \in Y$ disjoint open neighbourhoods $U_{x,y} \supset \{x\}$ and $U_y \supset \{y\}$. By the nature of the subspace topology of $Y$, the restriction of all the $U_y$ to $Y$ is an open cover of $Y$:
Now by the assumption that $Y$ is compact, there exists a finite subcover, hence a finite set $S \subset Y$ such that
is still a cover.
But the finite intersection
of the corresponding open neighbourhoods of $x$ is still open, and by construction it is disjoint from all the $U_{s}$, hence also from their union
Therefore $U_x$ and $U_Y$ are two open subsets as required.
Created on May 19, 2017 at 10:10:29. See the history of this page for a list of all contributions to it.