nLab
locally compact groupoid
Contents
Context
Topology
topology (point-set topology , point-free topology )

see also differential topology , algebraic topology , functional analysis and topological homotopy theory

Introduction

Basic concepts

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Category theory
Contents
Idea
There are a number of slightly different definitions in the literature, this page will try to give a general overarching definition.

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.

Examples
References
An early reference is

Jean Renault, A Groupoid Approach to C-Algebras_, Lecture Notes in Mathematics 793 (1980) doi:10.1007/BFb0091072
A version where Hausdorffness is weakened so that only the space of objects is Hausdorff is

Jean-Louis Tu, Non-Hausdorff groupoids, proper actions and K-theory , Documenta Mathematica, Vol. 9 (2004) pp565-597, journal page , arXiv:math/0403071
Last revised on February 8, 2019 at 05:44:23.
See the history of this page for a list of all contributions to it.