Continuous convergence in C(X)

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

Vol. 52 (1974), No. 2, 457–465

DOI: 10.2140/pjm.1974.52.457

Abstract

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

Milestones

Received: 7 September 1973

Revised: 6 March 1974

Published: 1 June 1974