Abstract
The second author and Smith proved that the product of two ordinals is hereditarily countably metacompact [5]. It is natural to ask whether X×Y is countably metacompact for every LOTS' X and Y. We answer the problem negatively, in fact, for every regular uncountable cardinal κ, we construct a hereditarily paracompact LOTS Lκ such that Lκ×S is not countably metacompact for any stationary set S in κ. Moreover we will find a condition on a GO-space X in order that X×κ is countably metacompact. As a corollary, we see that a subspace X of an ordinal is paracompact iff X×Y is countably metacompact for every GO-space Y.
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