Displaying similar documents to “A β -normal Tychonoff space which is not normal”

G δ -modification of compacta and cardinal invariants

Aleksander V. Arhangel'skii (2006)

Commentationes Mathematicae Universitatis Carolinae

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Given a space X , its G δ -subsets form a basis of a new space X ω , called the G δ -modification of X . We study how the assumption that the G δ -modification X ω is homogeneous influences properties of X . If X is first countable, then X ω is discrete and, hence, homogeneous. Thus, X ω is much more often homogeneous than X itself. We prove that if X is a compact Hausdorff space of countable tightness such that the G δ -modification of X is homogeneous, then the weight w ( X ) of X does not exceed 2 ω (Theorem 1)....

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...

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...