Products of spaces of ordinal numbers

The countable uniform power (or uniform box product) of a uniform space X is a special topology on ω X that lies between the Tychonoff topology and the box topology. We solve an open problem posed by P. Nyikos showing that if X is a compact proximal space then the countable uniform power of X is also proximal (although it is not compact). By recent results of J. R. Bell and G. Gruenhage this implies that the countable uniform power of a Corson compactum is collectionwise normal, countably paracompact and Fréchet-Urysohn. We also give some results about first countability, realcompactness in countable uniform powers of compact spaces and explore questions by P. Nyikos about semi-proximal spaces.

Section: Lemma Let a Bmentioning

confidence: 61%

“…In [7], normality-type properties of products of subspaces of ordinals were studied. In Corollary 3.3 of that paper, the authors prove that a product A × B, where A, B ⊂ ω 1 , is normal if and only if either one of A or B is not stationary or A ∩ B is stationary.…”

Section: Theoremmentioning

confidence: 99%

Uniform powers of compacta and the proximal game

Hernández-Gutiérrez

Szeptycki

2017

“…Moreover, using elementary submodels, it is proved in [4] that products of arbitrary many ordinals are mildly normal. In [6], it is proved that for A, B ⊆ ω 1 , A × B is normal if and only if A or B is non-stationary or A ∩ B is stationary in ω 1 . Since there are disjoint stationary sets A and B in ω 1 , there is a non-normal product A × B of two subspaces of ω 1 .…”

Section: Introductionmentioning

confidence: 98%

Mild normality of finite products of subspaces of ω1

Hirata¹,

Kemoto²

2006

Self Cite

It is known that products of arbitrary many ordinals are mildly normal [L. Kalantan, P.J. Szeptycki, κ-normality and products of ordinals, Topology Appl. 123 (2002) 537-545] and products of two subspaces of ordinals are also mildly normal [L. Kalantan, N. Kemoto, Mild normality in products of ordinals, Houston J. Math. 29 (2003) 937-947]. It was asked if products of arbitrary many subspaces of ordinals are mildly normal. In this paper, we characterize the mild normality of products of finitely many subspaces of ω 1 . Using this characterization, we show that there exist 3 subspaces of ω 1 whose product is not mildly normal.

“…• ω 1 × (ω 1 + 1) is countably paracompact but not normal, • if A and B are disjoint stationary sets in ω 1 , then A ×B is neither normal nor countably paracompact [11], • ω ω 1 is normal, but ω 1 ω 1 is not normal [3], • κ ω 1 is countably (para)compact for every κ,…”

Section: Introductionmentioning

confidence: 99%

Topological properties of products of ordinals

Kemoto

Szeptycki

2004

Self Cite

We study separation and covering properties of special subspaces of products of ordinals. In particular, it is proven that certain subspaces of Σ-products of ordinals are quasi-perfect preimages of Σ-products of copies of ω. We obtain as corollaries that products of ordinals are κ-normal and strongly zero-dimensional. Also, σ -products and Σ-products of ordinals are shown to be countably paracompact, κ-normal and strongly zero-dimensional. Normality in Σ-products and σ -products of ordinals is also characterized. It is also shown that any continuous real-valued function on a σ -product of ordinals has countable range.