Displaying similar documents to “ 𝒯 0 - and 𝒯 1 -reflections”

Condensations of Cartesian products

Oleg I. Pavlov (1999)

Commentationes Mathematicae Universitatis Carolinae

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We consider when one-to-one continuous mappings can improve normality-type and compactness-type properties of topological spaces. In particular, for any Tychonoff non-pseudocompact space X there is a μ such that X μ can be condensed onto a normal ( σ -compact) space if and only if there is no measurable cardinal. For any Tychonoff space X and any cardinal ν there is a Tychonoff space M which preserves many properties of X and such that any one-to-one continuous image of M μ , μ ≤ ν , contains a...

Linear extensions of relations between vector spaces

Árpád Száz (2003)

Commentationes Mathematicae Universitatis Carolinae

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Let X and Y be vector spaces over the same field K . Following the terminology of Richard Arens [Pacific J. Math. 11 (1961), 9–23], a relation F of X into Y is called linear if λ F ( x ) ⊂ F ( λ x ) and F ( x ) + F ( y ) ⊂ F ( x + y ) for all λ ∈ K ∖ { 0 } and x , y ∈ X . After improving and supplementing some former results on linear relations, we show that a relation Φ of a linearly independent subset E of X into Y can be extended to a linear relation F of X into Y if and only if there exists a linear subspace Z of Y such that Φ ( e ) ∈ Y | Z for all e ∈ E . Moreover, if...

A β -normal Tychonoff space which is not normal

Eva Murtinová (2002)

Commentationes Mathematicae Universitatis Carolinae

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α -normality and β -normality are properties generalizing normality of topological spaces. They consist in separating dense subsets of closed disjoint sets. We construct an example of a Tychonoff β -normal non-normal space and an example of a Hausdorff α -normal non-regular space.

On α -normal and β -normal spaces

Aleksander V. Arhangel'skii, Lewis D. Ludwig (2001)

Commentationes Mathematicae Universitatis Carolinae

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We define two natural normality type properties, α -normality and β -normality, and compare these notions to normality. A natural weakening of Jones Lemma immediately leads to generalizations of some important results on normal spaces. We observe that every β -normal, pseudocompact space is countably compact, and show that if X is a dense subspace of a product of metrizable spaces, then X is normal if and only if X is β -normal. All hereditarily separable spaces are α -normal. A space is...