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On the absoluteness of openly-generated and Dugundji spaces. (English) Zbl 0795.54031

The author discusses the preservation of certain topological properties. Given a pair of models of set-theory \(M \in V\) and a space \(X \in M\), such that \(M \models\) “\(X\) is compact” it is common to consider the topology on \(X\) in \(V\) which is generated by the topology in \(M\). The resulting space \(X\) is not, in general, compact in \(V\). The author defines a natural compactification of \(X\) – the smallest one such that all the continuous real-valued functions from \(M\) will extend continuously. The categorical properties of this assignment are studied (preserved under products, inverse limits) and some of the mapping properties are noted. In addition the connection with proximities is explored. Finally this machinery is applied to the class of Dugundji spaces in the case that \(V\) is obtained by a ccc forcing extension over \(M\).
Reviewer: A.Dow (North York)

MSC:

54C99 Maps and general types of topological spaces defined by maps
54A35 Consistency and independence results in general topology
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