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Products with linear and countable type factors. (English) Zbl 0917.54012

A space is said to be suborderable if it can be embedded in a linearly ordered topological space. Extending the first author’s theorem that the product of a suborderable space and a countable space is normal, it is shown that the product \(X \times Y\) is normal if (a) \(X\) is suborderable and (b) \(Y\) has countable tightness and \(Y \times Z\) is normal for any paracompact suborderable space \(Z\). For example, a space \(Y\) satisfies (b) above if it is a countably tight \(\sigma\)-compact space, or a countable tight, paracompact, \(\sigma\)-locally compact space. Suborderability of \(X\) cannot be replaced by monotone normality. Indeed, an example is given which shows that the product of a monotonically normal space and a countable space need not be normal. However it is proved that the product of a monotonically normal space and a monotonically normal countable space is normal.

MSC:

54B10 Product spaces in general topology
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54E18 \(p\)-spaces, \(M\)-spaces, \(\sigma\)-spaces, etc.
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
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References:

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