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

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

A function f:XVf\colon X \to V on a topological space with values in a vector space VV (or really any pointed set with the basepoint called 00) 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 f 1(V{0})¯\overline{f^{-1}(V \setminus \{0\})} is a compact subset of XX.

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

If we work with locales instead of topological spaces, then a closed point 00 in VV still has an open subspace of VV as its formal dual, and we use this in the place of V{0}V \setminus \{0\}.

Revised on November 2, 2017 15:41:28 by Urs Schreiber (46.183.103.8)