Countable metacompactness of products of LOTS'

The second author and Smith proved that the product of two ordinals is hereditarily countably metacompact [5]. It is natural to ask whether X×YX×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 LκLκ such that Lκ×SLκ×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×κX×κ is countably metacompact. As a corollary, we see that a subspace X   of an ordinal is paracompact iff X×YX×Y is countably metacompact for every GO-space Y.