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On the Compactness and Countable Compactness of 2 ℝ in ZF

Kyriakos Keremedis, Evangelos Felouzis, Eleftherios Tachtsis (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements " 2 ℝ is countably compact" and " 2 ℝ is compact"

On the extent of star countable spaces

Ofelia Alas, Lucia Junqueira, Jan Mill, Vladimir Tkachuk, Richard Wilson (2011)

Open Mathematics

For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ⊂ X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have...

On the Lindelöf property of spaces of continuous functions over a Tychonoff space and its subspaces

Oleg Okunev (2009)

Commentationes Mathematicae Universitatis Carolinae

We study relations between the Lindelöf property in the spaces of continuous functions with the topology of pointwise convergence over a Tychonoff space and over its subspaces. We prove, in particular, the following: a) if C p ( X ) is Lindelöf, Y = X ∪ { p } , and the point p has countable character in Y , then C p ( Y ) is Lindelöf; b) if Y is a cozero subspace of a Tychonoff space X , then l ( C p ( Y ) ω ) ≤ l ( C p ( X ) ω ) and ext ( C p ( Y ) ω ) ≤ ext ( C p ( X ) ω ) .

On the n -fold symmetric product of a space with a σ - ( P ) -property c n -network ( c k -network)

Luong Q. Tuyen, Ong V. Tuyen (2020)

Commentationes Mathematicae Universitatis Carolinae

We study the relation between a space X satisfying certain generalized metric properties and its n -fold symmetric product ℱ n ( X ) satisfying the same properties. We prove that X has a σ - ( P ) -property c n -network if and only if so does ℱ n ( X ) . Moreover, if X is regular then X has a σ - ( P ) -property c k -network if and only if so does ℱ n ( X ) . By these results, we obtain that X is strict σ -space (strict ℵ -space) if and only if so is ℱ n ( X ) .

On the paracompactness of frames

Xiangdong Chen (1992)

Commentationes Mathematicae Universitatis Carolinae

Through the study of frame congruences, new characterizations of the paracompactness of frames are obtained.

On weaker forms of the chain (F) condition and metacompactness-like covering properties in the product spaces

Süleyman Önal, Çetin Vural (2013)

Open Mathematics

We introduce the concept of a family of sets generating another family. Then we prove that if X is a topological space and X has W = {W(x): x ∈ X} which is finitely generated by a countable family satisfying (F) which consists of families each Noetherian of ω-rank, then X is metaLindelöf as well as a countable product of them. We also prove that if W satisfies ω-rank (F) and, for every x ∈ X, W(x) is of the form W 0(x) ∪ W 1(x), where W 0(x) is Noetherian and W 1(x) consists of neighbourhoods of...

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