Vol. 52, No. 2, 1974

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Continuous convergence in C(X)

Darrell Conley Kent, Kelly Denis McKennon, G. Richardson and M. Schroder

Vol. 52 (1974), No. 2, 457–465

Let X be a convergence space and C(X) the R-algebra of all continuous real-valued functions on X, equipped with the continuous convergence structure. If the natural map from X into C(C(X)) is an embedding, then X is said to be a c-space. With each space X there is associated the c-modification cX which is a c-space with the property C(X) = C(cX). This leads to the following theorems which are valid for any convergence space X: (1) C(X) is a topological space iff cX is locally compact; (2) C(X) is locally compact iff cX is finite.

Mathematical Subject Classification 2000
Primary: 54C35
Secondary: 54A20
Received: 7 September 1973
Revised: 6 March 1974
Published: 1 June 1974
Darrell Conley Kent
Kelly Denis McKennon
G. Richardson
M. Schroder