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Restricted compactness properties and their preservation under products. (English) Zbl 0962.54020

The topological space \(X\) is said to be CLP-compact if every clopen cover of \(X\) has a finite subcover (i.e., the zero-dimensional modification \(zX\) is compact). It is proved that a product of two CLP-compact spaces \(X,Y\) is CLP-compact iff \(z(X\times Y)=zX\times zY\) iff \(pr_X:X\times Y\to X\) is clopen. Under some weight restrictions, a product of two CLP-compact spaces is CLP-compact. There are three CLP-compact spaces the product of which is not CLP-compact but any 2-subproduct is CLP-compact. There is a model of set-theory containing a CLP-compact and Hausdorff space the square of which is not CLP-compact. Other \(\mathcal P\)-compactness is considered in addition to \({\mathcal P}= \text{CLP}\) (= all clopen sets), mainly when \(\mathcal P\) consists of open sets having small boundaries. For instance, there exists a 2-dimensional subspace \(X\) of a Euclidean space such that \(X\) is compact and Hausdorff in the topology generated by open sets with finite boundaries (usual examples of such \(\mathcal P\)-compact and \(\mathcal P\)-Hausdorff spaces are at most 1-dimensional).

MSC:

54D30 Compactness
54B10 Product spaces in general topology
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54F45 Dimension theory in general topology
54A35 Consistency and independence results in general topology
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