Displaying similar documents to “On n -thin dense sets in powers of topological spaces”

The "Full Clarkson-Erdős-Schwartz Theorem" on the closure of non-dense Müntz spaces

Tamás Erdélyi (2003)

Studia Mathematica

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Denote by spanf₁,f₂,... the collection of all finite linear combinations of the functions f₁,f₂,... over ℝ. The principal result of the paper is the following. Theorem (Full Clarkson-Erdős-Schwartz Theorem). Suppose ( λ j ) j = 1 is a sequence of distinct positive numbers. Then s p a n 1 , x λ , x λ , . . . is dense in C[0,1] if and only if j = 1 ( λ j ) / ( λ j ² + 1 ) = . Moreover, if j = 1 ( λ j ) / ( λ j ² + 1 ) < , then every function from the C[0,1] closure of s p a n 1 , x λ , x λ , . . . can be represented as an analytic function on z ∈ ℂ ∖ (-∞, 0]: |z| < 1 restricted to (0,1). This result improves an...

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 .

A countably cellular topological group all of whose countable subsets are closed need not be -factorizable

Mihail G. Tkachenko (2023)

Commentationes Mathematicae Universitatis Carolinae

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We construct a Hausdorff topological group G such that 1 is a precalibre of G (hence, G has countable cellularity), all countable subsets of G are closed and C -embedded in G , but G is not -factorizable. This solves Problem 8.6.3 from the book “Topological Groups and Related Structures" (2008) in the negative.

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.

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...

Finite-dimensional maps and dendrites with dense sets of end points

Hisao Kato, Eiichi Matsuhashi (2006)

Colloquium Mathematicae

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The first author has recently proved that if f: X → Y is a k-dimensional map between compacta and Y is p-dimensional (0 ≤ k, p < ∞), then for each 0 ≤ i ≤ p + k, the set of maps g in the space C ( X , I p + 2 k + 1 - i ) such that the diagonal product f × g : X Y × I p + 2 k + 1 - i is an (i+1)-to-1 map is a dense G δ -subset of C ( X , I p + 2 k + 1 - i ) . In this paper, we prove that if f: X → Y is as above and D j (j = 1,..., k) are superdendrites, then the set of maps h in C ( X , j = 1 k D j × I p + 1 - i ) such that f × h : X Y × ( j = 1 k D j × I p + 1 - i ) is (i+1)-to-1 is a dense G δ -subset of C ( X , j = 1 k D j × I p + 1 - i ) for each 0 ≤ i ≤ p.

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...

Coloring Cantor sets and resolvability of pseudocompact spaces

István Juhász, Lajos Soukup, Zoltán Szentmiklóssy (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let us denote by Φ ( λ , μ ) the statement that 𝔹 ( λ ) = D ( λ ) ω , i.e. the Baire space of weight λ , has a coloring with μ colors such that every homeomorphic copy of the Cantor set in 𝔹 ( λ ) picks up all the μ colors. We call a space X π -regular if it is Hausdorff and for every nonempty open set U in X there is a nonempty open set V such that V ¯ U . We recall that a space X is called feebly compact if every locally finite collection of open sets in X is finite. A Tychonov space is pseudocompact if and...

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...

A note on spaces with countable extent

Yan-Kui Song (2017)

Commentationes Mathematicae Universitatis Carolinae

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Let P be a topological property. A space X is said to be star P if whenever 𝒰 is an open cover of X , there exists a subspace A X with property P such that X = S t ( A , 𝒰 ) . In this note, we construct a Tychonoff pseudocompact SCE-space which is not star Lindelöf, which gives a negative answer to a question of Rojas-Sánchez and Tamariz-Mascarúa.

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 .

On non-normality points, Tychonoff products and Suslin number

Sergei Logunov (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let a space X be Tychonoff product α < τ X α of τ -many Tychonoff nonsingle point spaces X α . Let Suslin number of X be strictly less than the cofinality of τ . Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification β X . In particular, this is true if X is either R τ or ω τ and a cardinal τ is infinite and not countably cofinal.