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Borsuk-Sieklucki theorem in cohomological dimension theory

Margareta Boege, Jerzy Dydak, Rolando Jiménez, Akira Koyama, Evgeny V. Shchepin (2002)

Fundamenta Mathematicae

The Borsuk-Sieklucki theorem says that for every uncountable family X α α ∈ A of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that d i m ( X α ∩ X β ) = n . In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is c l c ℤ n + 1 , where n ≥ 1, and G is an Abelian group. Let X α α ∈ J be an uncountable family of closed subsets of X. If d i m G X = d i m G X α = n for all α ∈ J, then d i m G ( X α ∩ X β ) = n for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski...

C p ( I ) is not subsequential

Viacheslav I. Malykhin (1999)

Commentationes Mathematicae Universitatis Carolinae

If a separable dense in itself metric space is not a union of countably many nowhere dense subsets, then its C p -space is not subsequential.

C ( X ) can sometimes determine X without X being realcompact

Melvin Henriksen, Biswajit Mitra (2005)

Commentationes Mathematicae Universitatis Carolinae

As usual C ( X ) will denote the ring of real-valued continuous functions on a Tychonoff space X . It is well-known that if X and Y are realcompact spaces such that C ( X ) and C ( Y ) are isomorphic, then X and Y are homeomorphic; that is C ( X ) determines X . The restriction to realcompact spaces stems from the fact that C ( X ) and C ( υ X ) are isomorphic, where υ X is the (Hewitt) realcompactification of X . In this note, a class of locally compact spaces X that includes properly the class of locally compact realcompact spaces is exhibited...

Cardinal invariants of universals

Gareth Fairey, Paul Gartside, Andrew Marsh (2005)

Commentationes Mathematicae Universitatis Carolinae

We examine when a space X has a zero set universal parametrised by a metrisable space of minimal weight and show that this depends on the σ -weight of X when X is perfectly normal. We also show that if Y parametrises a zero set universal for X then h L ( X n ) ≤ h d ( Y ) for all n ∈ ℕ . We construct zero set universals that have nice properties (such as separability or ccc) in the case where the space has a K -coarser topology. Examples are given including an S space with zero set universal parametrised by an L space (and...

Cartesian closed hull for (quasi-)metric spaces (revisited)

Mark Nauwelaerts (2000)

Commentationes Mathematicae Universitatis Carolinae

An existing description of the cartesian closed topological hull of p MET ∞ , the category of extended pseudo-metric spaces and nonexpansive maps, is simplified, and as a result, this hull is shown to be a special instance of a “family” of cartesian closed topological subconstructs of p q s MET ∞ , the category of extended pseudo-quasi-semi-metric spaces (also known as quasi-distance spaces) and nonexpansive maps. Furthermore, another special instance of this family yields the cartesian closed topological hull of...

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