nLab support of a function

This article is about support of a function. For other notions of support, see 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

In set theory

Given a pointed set AA with specified element 0A0 \in A, a set XX, and a function f:XAf \colon X \to A, the support of ff is the subset of XX on which ff is not equal to 00.

In constructive mathematics

In constructive mathematics, there are multiple notions of inequality, due to the failure of the double negation law. As a result, there are multiple notion of support of a function. Thus, we define the following:

Given a pointed set AA with specified element 0A0 \in A, a set XX, and a function f:XAf \colon X \to A, the support of ff is the subset of XX on which ff is not equal to 00.

Given a pointed set AA with an tight apartness relation #\# and specified element 0A0 \in A, a set XX, and a function f:XAf \colon X \to A, the strong support of ff is the subset of XX on which ff is apart from 00.

In topology

In topology the support of a continuous function f:XAf \colon X \to A as above is the topological closure of the set of points on which ff does not vanish:

Supp(f)=Cl({xX|f(x)0A}). Supp(f) = Cl(\{x \in X \vert f(x) \neq 0 \in A\}) \,.

If Supp(f)XSupp(f) \subset X is a compact subspace, then one says that ff has compact support.

See also

References

Last revised on October 17, 2022 at 16:52:47. See the history of this page for a list of all contributions to it.