Displaying similar documents to “About w c s -covers and w c s * -networks on the Vietoris hyperspace ( X )

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

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

Locally functionally countable subalgebra of ( L )

M. Elyasi, A. A. Estaji, M. Robat Sarpoushi (2020)

Archivum Mathematicum

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Let L c ( X ) = { f C ( X ) : C f ¯ = X } , where C f is the union of all open subsets U X such that | f ( U ) | 0 . In this paper, we present a pointfree topology version of L c ( X ) , named c ( L ) . We observe that c ( L ) enjoys most of the important properties shared by ( L ) and c ( L ) , where c ( L ) is the pointfree version of all continuous functions of C ( X ) with countable image. The interrelation between ( L ) , c ( L ) , and c ( L ) is examined. We show that L c ( X ) c ( 𝔒 ( X ) ) for any space X . Frames L for which c ( L ) = ( L ) are characterized.

Seeking a network characterization of Corson compacta

Ziqin Feng (2021)

Commentationes Mathematicae Universitatis Carolinae

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We say that a collection 𝒜 of subsets of X has property ( C C ) if there is a set D and point-countable collections 𝒞 of closed subsets of X such that for any A 𝒜 there is a finite subcollection of 𝒞 such that A = D . Then we prove that any compact space is Corson if and only if it has a point- σ - ( C C ) base. A characterization of Corson compacta in terms of (strong) point network is also given. This provides an answer to an open question in “A Biased View of Topology as a Tool in Functional Analysis”...

Spaces with property ( D C ( ω 1 ) )

Wei-Feng Xuan, Wei-Xue Shi (2017)

Commentationes Mathematicae Universitatis Carolinae

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We prove that if X is a first countable space with property ( D C ( ω 1 ) ) and with a G δ -diagonal then the cardinality of X is at most 𝔠 . We also show that if X is a first countable, DCCC, normal space then the extent of X is at most 𝔠 .

Functionally countable subalgebras and some properties of the Banaschewski compactification

A. R. Olfati (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let X be a zero-dimensional space and C c ( X ) be the set of all continuous real valued functions on X with countable image. In this article we denote by C c K ( X ) (resp., C c ψ ( X ) ) the set of all functions in C c ( X ) with compact (resp., pseudocompact) support. First, we observe that C c K ( X ) = O c β 0 X X (resp., C c ψ ( X ) = M c β 0 X υ 0 X ), where β 0 X is the Banaschewski compactification of X and υ 0 X is the -compactification of X . This implies that for an -compact space X , the intersection of all free maximal ideals in C c ( X ) is equal to C c K ( X ) , i.e., M c β 0 X X = C c K ( X ) . By applying...

Σ s -products revisited

Reynaldo Rojas-Hernández (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that any Σ s -product of at most 𝔠 -many L Σ ( ω ) -spaces has the L Σ ( ω ) -property. This result generalizes some known results about L Σ ( ω ) -spaces. On the other hand, we prove that every Σ s -product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every Σ s -product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

On almost everywhere differentiability of the metric projection on closed sets in l p ( n ) , 2 < p <

Tord Sjödin (2018)

Czechoslovak Mathematical Journal

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Let F be a closed subset of n and let P ( x ) denote the metric projection (closest point mapping) of x n onto F in l p -norm. A classical result of Asplund states that P is (Fréchet) differentiable almost everywhere (a.e.) in n in the Euclidean case p = 2 . We consider the case 2 < p < and prove that the i th component P i ( x ) of P ( x ) is differentiable a.e. if P i ( x ) x i and satisfies Hölder condition of order 1 / ( p - 1 ) if P i ( x ) = x i .

On hereditary normality of ω * , Kunen points and character ω 1

Sergei Logunov (2021)

Commentationes Mathematicae Universitatis Carolinae

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We show that ω * { p } is not normal, if p is a limit point of some countable subset of ω * , consisting of points of character ω 1 . Moreover, such a point p is a Kunen point and a super Kunen point.

On subcompactness and countable subcompactness of metrizable spaces in ZF

Kyriakos Keremedis (2022)

Commentationes Mathematicae Universitatis Carolinae

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We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space 𝐗 = ( X , T ) is countably compact if and only if it is countably subcompact relative to T . (iii) For every metrizable space 𝐗 = ( X , T ) , the following are equivalent: (a) 𝐗 is compact; (b) for every open filter of 𝐗 , { F ¯ : F } ; (c) 𝐗 is subcompact relative to T . We also show: (iv) The negation of each of the statements, (a) every countably subcompact...

Selectors of discrete coarse spaces

Igor Protasov (2022)

Commentationes Mathematicae Universitatis Carolinae

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Given a coarse space ( X , ) with the bornology of bounded subsets, we extend the coarse structure from X × X to the natural coarse structure on ( { } ) × ( { } ) and say that a macro-uniform mapping f : ( { } ) X (or f : [ X ] 2 X ) is a selector (or 2-selector) of ( X , ) if f ( A ) A for each A { } ( A [ X ] 2 , respectively). We prove that a discrete coarse space ( X , ) admits a selector if and only if ( X , ) admits a 2-selector if and only if there exists a linear order “ " on X such that the family of intervals { [ a , b ] : a , b X , a b } is a base for the bornology .