TheoremBase

The Space of Continuous Functions with Compact Support on a Locally Compact Separable Metric Space is Separable

For a locally compact separable metric space X, the real normed space of continuous compactly supported functions on X with the uniform norm is separable.

Statement

In the setting of Locally Compact Metric Spaces and Compactly Supported Functions: Standing Notation, let (X,d)(X,d) be a locally compact metric space that is separable, and let Cc(X)C_{c}(X) carry the uniform norm ∥⋅∥u\|\cdot\|_{u}.

Separability. (Cc(X),∥⋅∥u)(C_{c}(X),\|\cdot\|_{u}) is separable.

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