nLab
compact support
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

Contents
Definition
A function $f\colon X \to V$ on a topological space with values in a vector space $V$ (or really any pointed set with the basepoint called $0$ ) has compact support if the closure of its support , the set of points where it is non-zero, is a compact subset . That is, the subset $\overline{f^{-1}(V \setminus \{0\})}$ is a compact subset of $X$ .

Typically, $X$ is Hausdorff , $f$ is a continuous function , and $V$ is a Hausdorff topological vector space (or at least a pointed topological space whose basepoint is closed), so that $f^{-1}(V \setminus \{0\})$ is an open subspace of $X$ , yet any compact subspace of $X$ must be closed ; this is why we take the closure.

If we work with locales instead of topological spaces, then a closed point $0$ in $V$ still has an open subspace of $V$ as its formal dual, and we use this in the place of $V \setminus \{0\}$ .

Revised on May 12, 2017 08:17:02
by

Urs Schreiber
(92.218.150.85)