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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.
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