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A note on condensations of \(C_p(X)\) onto compacta. (English) Zbl 1090.54003

A condensation is a continuous bijection. The authors study when a space \(C_p(X)\) admits a condensation onto a compact Hausdorff space. Using clever combinations of a great number of known results they obtain several necessary conditions on \(X\). In particular \(C_p(X)\) admits no condensation onto a compact Hausdorff space if \(X\) is a non-metrizable Corson compactum. On the other hand, \(C_p(X)\) admits such a condensation if \(X\) is the double-arrow space. They also formulate some open problems. For example: Does \(C_p(\beta \omega )\) admit a condensation onto a compact Hausdorff space? Is it true that \(C_p(X)\) admits no condensation onto a \(\sigma \)-compact Tikhonov space if \(X\) is a non-metrizable Corson compact space?

MSC:

54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54C35 Function spaces in general topology
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