Paracompactness of lexicographic products of GO-spaces

Countable compactness of lexicographic products of GO-spaces

“…In this section, we characterize countable 0-compactness of lexicographic products of two GO-spaces. The following is easy to prove, see also [7,Lemma 3.6 (3a)].…”

“…Some of the definitions above are introduced in [7]. Note that 0 / ∈ L + ∪ L − and for an ordinal α ≥ ω, l(α) is the largest limit ordinal less than or equal to α, therefore the half open interval [l(α), α) of ordinals is finite.…”

“…(2) ⇒ (1) Assuming ( 2) and the negation of (1), take a closed 0-segment A of X with 0-cf X A = ω. Modifying the proof of Theorem 4.8 in [7], we consider 3 cases and their subcases. In each case, we will derive a contradiction.…”

“…The lexicographic product ω 1 × [0, 1) R is called the long line of length ω 1 and denoted by L(ω 1 ). Then it is known that ω 1 × S is paracompact but S × ω 1 is not paracompact, see [7]. On the other hand, both lexicographic products ω 1 × S and S × ω 1 are not countably compact, because S is not boundedly countably 0-compact.…”

“…Recently, the author defined the notion of the lexicographic product of GOspaces and extended the results above for GO-spaces, see [6], [7]. It is also known:…”