A Corson Compact Space is Countable if the Complement of its Diagonal is Functionally Countable

2021 ◽  
Vol 58 (3) ◽  
pp. 398-407
Author(s):  
Vladimir V. Tkachuk
Keyword(s):  
Continuous Function ◽  
Compact Space ◽  
Infinite Cardinal ◽  
Scattered Space ◽  
Corson Compact

A space X is called functionally countable if ƒ (X) is countable for any continuous function ƒ : X → Ø. Given an infinite cardinal k, we prove that a compact scattered space K with d(K) > k must have a convergent k+-sequence. This result implies that a Corson compact space K is countable if the space (K × K) \ ΔK is functionally countable; here ΔK = {(x, x): x ϵ K} is the diagonal of K. We also establish that, under Jensen’s Axiom ♦, there exists a compact hereditarily separable non-metrizable compact space X such that (X × X) \ ΔX is functionally countable and show in ZFC that there exists a non-separable σ-compact space X such that (X × X) \ ΔX is functionally countable.

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 Fuzzy pre-continuous and fuzzy pre*-continuous function of fuzzy pre-compact space in fuzzy topological space

2015 ◽  
Vol 3 (1) ◽  
pp. 87
Author(s):  
Munir Abdul Khalik Al-Khafaji ◽  
Marwah Hasan
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Compact Space ◽  
Fuzzy Topological Space

<p>The aim of this paper is to introduce and study the notion of a fuzzy pre-continuous function, fuzzy pre*- continuous function, fuzzy pre-compact space and some properties, remarks related to them.</p>

Continuous Images of Compact Semilattices

1987 ◽  
Vol 30 (1) ◽  
pp. 109-113 ◽  
Author(s):  
Murray Bell ◽  
Jan Pelant
Keyword(s):  
Compact Space ◽  
Infinite Cardinal ◽  
Theoretic Property ◽  
Compact Spaces ◽  
General Order ◽  
Isolated Point ◽  
Continuous Images

AbstractHyadic spaces are the continuous images of a hyperspace of a compact space. We prove that every non-isolated point in a hyadic space is the endpoint of some infinite cardinal subspace. We isolate a more general order-theoretic property of hyerspaces of compact spaces which is also enjoyed by compact semilattices from which the theorem follows.

Unisolvence on Multidimensional Spaces

1968 ◽  
Vol 11 (3) ◽  
pp. 469-474 ◽  
Author(s):  
Charles B. Dunham
Keyword(s):  
Continuous Function ◽  
Euclidean Space ◽  
Compact Subset ◽  
Compact Space ◽  
General Condition ◽  
Chebyshev Approximation ◽  
Continuous Functions ◽  
Dimensional Euclidean Space ◽  
Variable Degree ◽  
Convexity Conditions

In this note we consider the possibility of unisolvence of a family of real continuous functions on a compact subset X of m-dimensional Euclidean space. Such a study is of interest for two reasons. First, an elegant theory of Chebyshev approximation has been constructed for the case where the approximating family is unisolvent of degree n on an interval [α, β]. We study what sort of theory results from unisolvence of degree n on a more general space. Secondly, uniqueness of best Chebyshev approximation on a general compact space to any continuous function on X can be shown if the approximating family is unisolvent of degree n and satisfies certain convexity conditions. It is therefore of importance to Chebyshev approximation to consider the domains X on which unisolvence can occur. We will also study a more general condition on involving a variable degree.

scholarly journals Extensions of best approximation and coincidence theorems

1997 ◽  
Vol 20 (4) ◽  
pp. 689-698 ◽  
Author(s):  
Sehie Park
Keyword(s):  
Continuous Function ◽  
Vector Space ◽  
Topological Vector Space ◽  
Compact Space ◽  
Best Approximation ◽  
Upper Semicontinuous ◽  
Coincidence Theorems ◽  
Continuous Seminorm ◽  
Upper Semicontinuous Multifunction

LetXbe a Hausdorff compact space,Ea topological vector space on whichE*separates points,F:X→2Ean upper semicontinuous multifunction with compact acyclic values, andg:X→Ea continuous function such thatg(X)is convex andg−1(y)is acyclic for eachy∈g(X). Then either (1) there exists anx0∈Xsuch thatgx0∈Fx0or (2) there exist an(x0,z0)on the graph ofFand a continuous seminormponEsuch that0<p(gx0−z0)≤p(y−z0)         for all         y∈g(X). A generalization of this result and its application to coincidence theorems are obtained. Our aim in this paper is to unify and improve almost fifty known theorems of others.

scholarly journals ON A PROBLEM OF TALAGRAND CONCERNING SEPARATELY CONTINUOUS FUNCTIONS

2020 ◽  
pp. 1-10
Author(s):  
Volodymyr Mykhaylyuk ◽  
Roman Pol
Keyword(s):  
Continuous Function ◽  
Compact Space ◽  
Continuous Functions ◽  
Baire Space ◽  
Negative Solution ◽  
Borel Set ◽  
Joint Continuity

We construct a separately continuous function $e:E\times K\rightarrow \{0,1\}$ on the product of a Baire space $E$ and a compact space $K$ such that no restriction of $e$ to any non-meagre Borel set in $E\times K$ is continuous. The function $e$ has no points of joint continuity, and, hence, it provides a negative solution of Talagrand’s problem in Talagrand [Espaces de Baire et espaces de Namioka, Math. Ann.270 (1985), 159–164].

scholarly journals Namioka spaces and topological games

2006 ◽  
Vol 73 (2) ◽  
pp. 263-272 ◽  
Author(s):  
V. V. Mykhaylyuk
Keyword(s):  
Continuous Function ◽  
Compact Space

We introduce a class of β − v-unfavourable spaces, which contains some known classes of β-unfavourable spaces for topological games of Choquet type. It is proved that every β − v-unfavourable space X is a Namioka space, that is for any compact space Y and any separately continuous function f : x × Y → ℝ there exists a dense in XGδ-set A ⊆ X such that f is jointly continuous at each point of A × Y.

scholarly journals Every Radon-Nikodym Corson compact space is Eberlein compact

1991 ◽  
Vol 98 (2) ◽  
pp. 157-174 ◽  
Author(s):  
J. Orihuela ◽  
W. Schachermayer ◽  
M. Valdivia
Keyword(s):  
Compact Space ◽  
Eberlein Compact ◽  
Corson Compact

scholarly journals Locally compact space and continuity

BIBECHANA ◽  
1970 ◽  
Vol 7 ◽  
pp. 18-20
Author(s):  
Shitanshu Shekhar Choudhary ◽  
Raju Ram Thapa
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Compact Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Locally Compact ◽  
Locally Compact Space

Topological spaces for being T0, T1, T2 and regular space have been discussed. The conditions for a topological space to be locally compact have also been studied. We have found that a continuous function preserves locally compactness. Keywords: Topological spaces; Compactness; Regular space DOI: 10.3126/bibechana.v7i0.4038BIBECHANA 7 (2011) 18-20

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