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1987•Proceedings of the American Mathematical SocietyOpen access

Approximate identities and paracompactness

Robert A. Fontenot, Robert F. Wheeler

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Abstract

Let $X$ denote a locally compact Hausdorff space and ${C_b}(X)$ the algebra of continuous complex-valued functions on $X$. The main result of this paper is that $X$ is paracompact if and only if ${C_0}(X)$, the subalgebra of ${C_b}(X)$ consisting of functions which vanish at infinity, has an approximate identity which is a relatively compact subset of ${C_b}(X)$ for the weak topology of the pairing of ${C_b}(X)$ with its strict topology dual.

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Let $X$ denote a locally compact Hausdorff space and ${C_b}(X)$ the algebra of continuous complex-valued functions on $X$. The main result of this paper is that $X$ is paracompact if and only if ${C_0}(X)$, the subalgebra of ${C_b}(X)$ consisting of functions which vanish at infinity, has an approximate identity which is a relatively compact subset of ${C_b}(X)$ for the weak topology of the pairing of ${C_b}(X)$ with its strict topology dual.

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Available abstract

Let $X$ denote a locally compact Hausdorff space and ${C_b}(X)$ the algebra of continuous complex-valued functions on $X$. The main result of this paper is that $X$ is paracompact if and only if ${C_0}(X)$, the subalgebra of ${C_b}(X)$ consisting of functions which vanish at infinity, has an approximate identity which is a relatively compact subset of ${C_b}(X)$ for the weak topology of the pairing of ${C_b}(X)$ with its strict topology dual.

Key concepts: Paracompact space, Hausdorff space, Locally compact space, Mathematics, Subalgebra, Continuous functions on a compact Hausdorff space, Dual space, Topology (electrical circuits)

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