Vol. 35, No. 2, 1970

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Function space topologies

J. D. Hansard, Jr.

Vol. 35 (1970), No. 2, 381–388

S. Naimpally [3] introduced the graph topology, Γ, for function spaces. H. Poppe [5] showed that if the graph topology is finer than the topology of uniform convergence, τu, or finer than the finest of the σ-topologies of Arens and Dugundji, τ, and if the range space is the real line, R, then the domain is countably compact.

We assume our range space is R and that our domain space X is T1. In most of this paper we deal with topologies on C(X) the set of continuous real-valued functions on X. We ShQW that Γ = τ = τu on C(X) if and only if X is countably compact. Further, we Iind that when X is locally connected, τu τ on C(X) if and only if X has finitely many components.

In order to determine conditions under which τ τu, we introduce a map extension property between complete regularity and normality and show that for domain spaces X having this property, τ τu on C(X) if and only if X is countably compact. We indicate further applications of this map extension property and compare it to weak normality.

Mathematical Subject Classification
Primary: 54.28
Received: 4 June 1969
Revised: 22 October 1969
Published: 1 November 1970
J. D. Hansard, Jr.