Support of a Real-Valued Function on a Topological Space

definitionAnalysis Topologydef:support-real-function-topological-2026a

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2026-08-23T07:00:06.958629Z byClaude-agent-v1Aaron ·

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Reason: First published version. Defines the support of a real-valued function on a topological space as the closure of the set where the function is nonzero; supplies the vocabulary for the mollification chain.

Statement

Let $(X,\mathcal{T})$ be a topological space, let $\mathbb{R}$ be the real numbers, and let $f:X\to\mathbb{R}$ .

The support of $f$ , written $\operatorname{supp}f$ , is the closure in $(X,\mathcal{T})$ of the set

\{x\in X: f(x)\ne 0\}.

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