Products of monotonically normal spaces with various special factors

Hirata, Yasushi; Kemoto, Nobuyuki; Yajima, Yukinobu

doi:10.1016/j.topol.2013.12.006

Cited by 8 publications

(1 citation statement)

References 12 publications

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“…In fact, it is known that if the product X × B of a GO-space X and a subspace B of an ordinal is normal, then X × B has the shrinking property [3,Corollaries 8.16,7.19,Theorem 7.11], in particular, X × B is countably metacompact. So it is natural to ask:…”

Section: Claim 6 Z Is Not Countably Metacompactmentioning

confidence: 99%

Countable metacompactness of products of LOTS'

Hirata

Kemoto

2014

Topology and its Applications

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Available online xxxx MSC: primary 54F05, 54D20, 54B10 secondary 03E10 Keywords: LOTS Countably metacompact Product space Subspace of an ordinal StationaryThe 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 L κ such that L κ × 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 .A topological space X is said to be countably metacompact if each countable open cover has a point finite open refinement. It is well-known that every LOTS is hereditarily countably metacompact. 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 L κ such that L κ × 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 .Spaces mean regular topological spaces having at least two points. Let < be a linear order on a set X. λ(<) denotes the usual order topology, that is, the topology generated by (a, →) : a ∈ X ∪ (←, b) : b ∈ X ✩

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Section: Claim 6 Z Is Not Countably Metacompactmentioning

confidence: 99%