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
There are a number of slightly different definitions in the literature, this page will try to give a general overarching definition.
A topological groupoid $X_1 \stackrel{\overset{s}{\to}}{\underset{t}{\to}} X_0$ is called locally compact if the maps $s$ and $t$ are locally proper open maps.
This is far more general than is usually assumed. One can additionally assume that either $X_1$ is locally compact or $X_0$ is locally compact, or that $X_0$ is Hausdorff or $X_1$ is locally Hausdorff, or that $X_1$ is ∞-compact
One usually wants to place a Haar system? on a locally compact groupoid. The obvious topological requirement of local compactness is further refined in order to make such a system of measures well-behaved.
An early reference is
A version where Hausdorffness is weakened so that only the space of objects is Hausdorff is
Last revised on February 8, 2019 at 00:44:23. See the history of this page for a list of all contributions to it.