The k-Extremally Disconnected Spaces as Projectives

1964 ◽  
Vol 16 ◽  
pp. 253-260 ◽  
Author(s):  
Henry B. Cohen
Keyword(s):  
Continuous Function ◽  
Boolean Algebra ◽  
Topological Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Dense Subset ◽  
Proper Subset ◽  
Stone Space ◽  
Infinite Cardinal ◽  
Extremally Disconnected

The letter k denotes an infinite cardinal. A space is a compact Hausdorff space unless otherwise indicated. A space is called extremally disconnected (k-extremally disconnected) if it is the Stone space for a complete (k-complete) Boolean algebra. A map is a continuous function from one space into another. A map f:X —> Y is called minimal if f is onto, but f(M) is properly contained in Y for each closed proper subset M of X. A space F is called free if F has a dense subset X such that every space-valued function on X extends to a map on all of F or, equivalently, if F is the Stone-Cech compactification of some discrete topological space X.

scholarly journals A GENERALIZATION OF SIERPINSKI THEOREM ON UNIQUE DETERMINING OF A SEPARATELY CONTINUOUS FUNCTION

2021 ◽  
Vol 9 (1) ◽  
pp. 250-263
Author(s):  
V. Mykhaylyuk ◽  
O. Karlova
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Baire Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Infinite Cardinal ◽  
Urysohn Space ◽  
Completely Regular

In 1932 Sierpi\'nski proved that every real-valued separately continuous function defined on the plane $\mathbb R^2$ is determined uniquely on any everywhere dense subset of $\mathbb R^2$. Namely, if two separately continuous functions coincide of an everywhere dense subset of $\mathbb R^2$, then they are equal at each point of the plane. Piotrowski and Wingler showed that above-mentioned results can be transferred to maps with values in completely regular spaces. They proved that if every separately continuous function $f:X\times Y\to \mathbb R$ is feebly continuous, then for every completely regular space $Z$ every separately continuous map defined on $X\times Y$ with values in $Z$ is determined uniquely on everywhere dense subset of $X\times Y$. Henriksen and Woods proved that for an infinite cardinal $\aleph$, an $\aleph^+$-Baire space $X$ and a topological space $Y$ with countable $\pi$-character every separately continuous function $f:X\times Y\to \mathbb R$ is also determined uniquely on everywhere dense subset of $X\times Y$. Later, Mykhaylyuk proved the same result for a Baire space $X$, a topological space $Y$ with countable $\pi$-character and Urysohn space $Z$. Moreover, it is natural to consider weaker conditions than separate continuity. The results in this direction were obtained by Volodymyr Maslyuchenko and Filipchuk. They proved that if $X$ is a Baire space, $Y$ is a topological space with countable $\pi$-character, $Z$ is Urysohn space, $A\subseteq X\times Y$ is everywhere dense set, $f:X\times Y\to Z$ and $g:X\times Y\to Z$ are weakly horizontally quasi-continuous, continuous with respect to the second variable, equi-feebly continuous wuth respect to the first one and such that $f|_A=g|_A$, then $f=g$. In this paper we generalize all of the results mentioned above. Moreover, we analize classes of topological spaces wich are favorable for Sierpi\'nsi-type theorems.

scholarly journals Gelfand theorem implies Stone representation theorem of Boolean rings

1995 ◽  
Vol 18 (4) ◽  
pp. 701-704
Author(s):  
Parfeny P. Saworotnow
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Representation Theorem ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Boolean Rings ◽  
Stone Representation ◽  
Stone Theorem

Stone Theorem about representing a Boolean algebra in terms of open-closed subsets of a topological space is a consequence of the Gelfand Theorem about representing aB∗- algebra as the algebra of continuous functions on a compact Hausdorff space.

scholarly journals On the theorem of Kishi for a continuous function-kernel

1974 ◽  
Vol 53 ◽  
pp. 127-135 ◽  
Author(s):  
Isao Higuchi ◽  
Masayuki Itô
Keyword(s):  
Continuous Function ◽  
Potential Theory ◽  
Compact Hausdorff Space ◽  
Symmetric Function ◽  
Hausdorff Space ◽  
Locally Compact ◽  
Continuity Principle ◽  
Lower Semi ◽  
Locally Compact Hausdorff Space

In the potential theory with respect to a non-symmetric function-kernel, the following theorem is obtained by M. Kishi ([3]).Let X be a locally compact Hausdorff space and G be a lower semi-continuous function-kernel on X. Assume that G(x, x)>0 for any x in X and that G and the adjoint kernel Ğ of G satisfy “the continuity principle”.

Zero-Dimensional Compactifications of Locally Compact Spaces

1974 ◽  
Vol 26 (4) ◽  
pp. 920-930 ◽  
Author(s):  
R. Grant Woods
Keyword(s):  
Topological Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Dense Subspace ◽  
Locally Compact ◽  
Partially Order ◽  
Compact Spaces ◽  
Hausdorff Topological Space ◽  
Continuous Map ◽  
Locally Compact Spaces

Let X be a locally compact Hausdorff topological space. A compactification of X is a compact Hausdorff space which contains X as a dense subspace. Two compactifications αX and γX of X are equivalent if there is a homeomorphism from αX onto γX that fixes X pointwise. We shall identify equivalent compactifications of a given space. If is a family of compactifications of X, we can partially order by saying that αX ≦ γX if there is a continuous map from γX onto αX that fixes X pointwise.

Power-collapsing games

2008 ◽  
Vol 73 (4) ◽  
pp. 1433-1457 ◽  
Author(s):  
Miloš S. Kurilić ◽  
Boris Šobot
Keyword(s):  
Boolean Algebra ◽  
Dense Subset ◽  
Winning Strategy ◽  
Zero Element ◽  
The Other ◽  
Complete Boolean Algebra ◽  
Infinite Cardinal ◽  
Other Hand ◽  
Game Theoretic

AbstractThe game is played on a complete Boolean algebra , by two players. White and Black, in κ-many moves (where κ is an infinite cardinal). At the beginning White chooses a non-zero element p ∈ . In the α-th move White chooses pα ∈ (0, p) and Black responds choosing iα ∈{0, 1}. White winsthe play iff . where and .The corresponding game theoretic properties of c.B.a.'s are investigated. So, Black has a winning strategy (w.s.) if κ ≥ π() or if contains a κ-closed dense subset. On the other hand, if White has a w.s., then κ ∈ . The existence of w.s. is characterized in a combinatorial way and in terms of forcing. In particular, if 2<κ = κ ∈ Reg and forcing by preserves the regularity of κ, then White has a w.s. iff the power 2κ is collapsed to κ in some extension. It is shown that, under the GCH, for each set S ⊆ Reg there is a c.B.a. such that White (respectively. Black) has a w.s. for each infinite cardinal κ ∈ S (resp. κ ∉ S). Also it is shown consistent that for each κ ∈ Reg there is a c.B.a. on which the game is undetermined.

A note on elementary equivalence of C(K) spaces

1987 ◽  
Vol 52 (2) ◽  
pp. 368-373 ◽  
Author(s):  
S. Heinrich ◽  
C. Ward Henson ◽  
L. C. Moore
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Supremum Norm ◽  
Closed Unit Ball ◽  
Elementary Equivalence ◽  
Bounded Number ◽  
Totally Disconnected

In this paper we give a closer analysis of the elementary properties of the Banach spaces C(K), where K is a totally disconnected, compact Hausdorff space, in terms of the Boolean algebra B(K) of clopen subsets of K. In particular we sharpen a result in [4] by showing that if B(K1) and B(K2) satisfy the same sentences with ≤ n alternations of quantifiers, then the same is true of C(K1) and C(K2). As a consequence we show that for each n there exist C(K) spaces which are elementarily equivalent for sentences with ≤ n quantifier alternations, but which are not elementary equivalent in the full sense. Thus the elementary properties of Banach spaces cannot be determined by looking at sentences with a bounded number of quantifier alternations.The notion of elementary equivalence for Banach spaces which is studied here was introduced by the second author [4] and is expressed using the language of positive bounded formulas in a first-order language for Banach spaces. As was shown in [4], two Banach spaces are elementarily equivalent in this sense if and only if they have isometrically isomorphic Banach space ultrapowers (or, equivalently, isometrically isomorphic nonstandard hulls.)We consider Banach spaces over the field of real numbers. If X is such a space, Bx will denote the closed unit ball of X, Bx = {x ϵ X∣ ∣∣x∣∣ ≤ 1}. Given a compact Hausdorff space K, we let C(K) denote the Banach space of all continuous real-valued functions on K, under the supremum norm. We will especially be concerned with such spaces when K is a totally disconnected compact Hausdorff space. In that case B(K) will denote the Boolean algebra of all clopen subsets of K. We adopt the standard notation from model theory and Banach space theory.

scholarly journals rc-continuous functions and functions with rc-strongly closed graph

2003 ◽  
Vol 2003 (72) ◽  
pp. 4547-4555
Author(s):  
Bassam Al-Nashef
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Continuous Functions ◽  
Closed Graph ◽  
Compact Spaces ◽  
Extremally Disconnected ◽  
The Family ◽  
Regular Closed

The family of regular closed subsets of a topological space is used to introduce two concepts concerning a functionffrom a spaceXto a spaceY. The first of them is the notion offbeing rc-continuous. One of the established results states that a spaceYis extremally disconnected if and only if each continuous function from a spaceXtoYis rc-continuous. The second concept studied is the notion of a functionfhaving an rc-strongly closed graph. Also one of the established results characterizes rc-compact spaces (≡S-closed spaces) in terms of functions that possess rc-strongly closed graph.

scholarly journals The Menger and projective Menger properties of function spaces with the set-open topology

2019 ◽  
Vol 69 (3) ◽  
pp. 699-706 ◽  
Author(s):  
Alexander V. Osipov
Keyword(s):  
Topological Space ◽  
Hausdorff Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Function Spaces ◽  
List Type ◽  
Finite Sets ◽  
Tychonoff Space ◽  
Menger Space

Abstract For a Tychonoff space X and a family λ of subsets of X, we denote by Cλ(X) the space of all real-valued continuous functions on X with the set-open topology. A Menger space is a topological space in which for every sequence of open covers 𝓤1, 𝓤2, … of the space there are finite sets 𝓕1 ⊂ 𝓤1, 𝓕2 ⊂ 𝓤2, … such that family 𝓕1 ∪ 𝓕2 ∪ … covers the space. In this paper, we study the Menger and projective Menger properties of a Hausdorff space Cλ(X). Our main results state that Cλ(X) is Menger if and only if Cλ(X) is σ-compact; Cp(Y | X) is projective Menger if and only if Cp(Y | X) is σ-pseudocompact where Y is a dense subset of X.

scholarly journals Compact and extremally disconnected spaces

2004 ◽  
Vol 2004 (20) ◽  
pp. 1047-1056
Author(s):  
Bhamini M. P. Nayar
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Closed Subset ◽  
Hausdorff Space ◽  
Hausdorff Spaces ◽  
Compact Spaces ◽  
Hausdorff Topological Space ◽  
Extremally Disconnected

Viglino defined a Hausdorff topological space to beC-compact if each closed subset of the space is anH-set in the sense of Veličko. In this paper, we study the class of Hausdorff spaces characterized by the property that each closed subset is anS-set in the sense of Dickman and Krystock. Such spaces are calledC-s-compact. Recently, the notion of strongly subclosed relation, introduced by Joseph, has been utilized to characterizeC-compact spaces as those with the property that each function from the space to a Hausdorff space with a strongly subclosed inverse is closed. Here, it is shown thatC-s-compact spaces are characterized by the property that each function from the space to a Hausdorff space with a strongly sub-semiclosed inverse is a closed function. It is established that this class of spaces is the same as the class of Hausdorff, compact, and extremally disconnected spaces. The class ofC-s-compact spaces is properly contained in the class ofC-compact spaces as well as in the class ofS-closed spaces of Thompson. In general, a compact space need not beC-s-compact. The product of twoC-s-compact spaces need not beC-s-compact.

scholarly journals A description of the Stone space of Banach latticeC(K,E)

2006 ◽  
Vol 4 (2) ◽  
pp. 221-224
Author(s):  
Zafer Ercan
Keyword(s):  
Banach Lattice ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Banach Lattices ◽  
Stone Space ◽  
Topological Description

We give a topological description of the Stone space ofC(K,E), Banach lattices of continuous functions from a compact Hausdorff spaceKinto a Banach latticeE.

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