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Addition theorems for dense subspaces

Aleksander V. Arhangel'skii (2015)

Commentationes Mathematicae Universitatis Carolinae

We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space X which is the union of two dense metrizable subspaces need not be a p -space. However, if a normal space X is the union of a finite family μ of dense subspaces each of which is metrizable by a complete metric, then X is also metrizable by...

Almost all submaximal groups are paracompact and σ-discrete

O. Alas, I. Protasov, M. Tkačenko, V. Tkachuk, R. Wilson, I. Yaschenko (1998)

Fundamenta Mathematicae

We prove that any topological group of a non-measurable cardinality is hereditarily paracompact and strongly σ-discrete as soon as it is submaximal. Consequently, such a group is zero-dimensional. Examples of uncountable maximal separable spaces are constructed in ZFC.

Almost closed sets and topologies they determine

Vladimir Vladimirovich Tkachuk, Ivan V. Yashchenko (2001)

Commentationes Mathematicae Universitatis Carolinae

We prove that every countably compact AP-space is Fréchet-Urysohn. It is also established that if X is a paracompact space and C p ( X ) is AP, then X is a Hurewicz space. We show that every scattered space is WAP and give an example of a hereditarily WAP-space which is not an AP-space.

Almost disjoint families and property (a)

Paul Szeptycki, Jerry Vaughan (1998)

Fundamenta Mathematicae

We consider the question: when does a Ψ-space satisfy property (a)? We show that if | A | < p then the Ψ-space Ψ(A) satisfies property (a), but in some Cohen models the negation of CH holds and every uncountable Ψ-space fails to satisfy property (a). We also show that in a model of Fleissner and Miller there exists a Ψ-space of cardinality p which has property (a). We extend a theorem of Matveev relating the existence of certain closed discrete subsets with the failure of property (a).

Bell-type inequalities for parametric families of triangular norms

Saskia Janssens, Bernard De Baets, Hans De Meyer (2004)

Kybernetika

In recent work we have shown that the reformulation of the classical Bell inequalities into the context of fuzzy probability calculus leads to related inequalities on the commutative conjunctor used for modelling pointwise fuzzy set intersection. Also, an important role has been attributed to commutative quasi-copulas. In this paper, we consider these new Bell-type inequalities for continuous t-norms. Our contribution is twofold: first, we prove that ordinal sums preserve these Bell-type inequalities;...

Calibres, compacta and diagonals

Paul Gartside, Jeremiah Morgan (2016)

Fundamenta Mathematicae

For a space Z let 𝒦(Z) denote the partially ordered set of all compact subspaces of Z under set inclusion. If X is a compact space, Δ is the diagonal in X², and 𝒦(X²∖Δ) has calibre (ω₁,ω), then X is metrizable. There is a compact space X such that X²∖Δ has relative calibre (ω₁,ω) in 𝒦(X²∖Δ), but which is not metrizable. Questions of Cascales et al. (2011) concerning order constraints on 𝒦(A) for every subspace of a space X are answered.

Cardinal inequalities implying maximal resolvability

Marek Balcerzak, Tomasz Natkaniec, Małgorzata Terepeta (2005)

Commentationes Mathematicae Universitatis Carolinae

We compare several conditions sufficient for maximal resolvability of topological spaces. We prove that a space X is maximally resolvable provided that for a dense set X 0 ⊂ X and for each x ∈ X 0 the π -character of X at x is not greater than the dispersion character of X . On the other hand, we show that this implication is not reversible even in the class of card-homogeneous spaces.

Cardinal invariants and compactifications

Anatoly A. Gryzlov (1994)

Commentationes Mathematicae Universitatis Carolinae

We prove that every compact space X is a Čech-Stone compactification of a normal subspace of cardinality at most d ( X ) t ( X ) , and some facts about cardinal invariants of compact spaces.

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