Classical extensions of the ring of continuous functions and the corresponding preimages of a completely regular space

A connection between the separability and the countable chain condition of spaces withL-property (a topological spaceXhasL-property if for every topological spaceY, separately continuous functionf:X×Y→ℝand open setI⊆ℝ,the setf−1(I)is anFσ-set) is studied. We show that every completely regular Baire space with theL-property and the countable chain condition is separable and constructs a nonseparable completely regular space with theL-property and the countable chain condition. This gives a negative answer to a question of M. Burke.

Some results involving weak compactness in C(X), C(υX) and C(X)′

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500015480 ◽

1975 ◽

Vol 19 (3) ◽

pp. 221-229 ◽

Cited By ~ 1

Author(s):

I. Tweddle

Keyword(s):

Dual Space ◽

Present Note ◽

Continuous Functions ◽

Weak Compactness ◽

Regular Space ◽

Compact Sets ◽

Completely Regular ◽

Countably Compact ◽

Countable Compactness

The main aim of the present note is to compare C(X) and C(υX), the spaces of real-valued continuous functions on a completely regular space X and its real 1–1 compactification υX, with regard to weak compactness and weak countable compactness. In a sense to be made precise below, it is shown that C(X) and C(υX) have the same absolutely convex weakly countably compact sets. In certain circumstances countable compactness may be replaced by compactness, in which case one obtains a nice representation of the Mackey completion of the dual space of C(X) (Theorems 5, 6, 7).

On -Cogenerated Commutative Unital -Algebras

Algebra ◽

10.1155/2013/452862 ◽

2013 ◽

Vol 2013 ◽

pp. 1-3 ◽

Cited By ~ 1

Author(s):

Ehsan Momtahan

Keyword(s):

Compact Space ◽

Commutative Algebra ◽

Cardinal Number ◽

Continuous Functions ◽

Regular Space ◽

Completely Regular ◽

Simple Characterization ◽

Maximal Ideals ◽

Complex Valued

Gelfand-Naimark's theorem states that every commutative -algebra is isomorphic to a complex valued algebra of continuous functions over a suitable compact space. We observe that for a completely regular space , is dense--separable if and only if is -cogenerated if and only if every family of maximal ideals of with zero intersection has a subfamily with cardinal number less than and zero intersection. This gives a simple characterization of -cogenerated commutative unital -algebras via their maximal ideals.

Weakly compact operators and the strict topologies

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700003270 ◽

1989 ◽

Vol 39 (3) ◽

pp. 353-359 ◽

Cited By ~ 6

Author(s):

José Aguayo ◽

José Sánchez

Keyword(s):

Banach Space ◽

Compact Operators ◽

Continuous Functions ◽

Supremum Norm ◽

Regular Space ◽

Weakly Compact ◽

Completely Regular ◽

Weakly Compact Operators ◽

Bounded Continuous Functions

Let X be a completely regular space. We denote by Cb(X) the Banach space of all real-valued bounded continuous functions on X endowed with the supremum-norm.In this paper we prove some characterisations of weakly compact operators defined from Cb(X) into a Banach space E which are continuous with respect to fit, βt, βr and βσ introduced by Sentilles.We also prove that (Cb,(X), βi), i = t, τσ , has the Dunford-Pettis property.

On approximation in weighted spaces of continuous vector-valued functions

Glasgow Mathematical Journal ◽

10.1017/s0017089500006662 ◽

1987 ◽

Vol 29 (1) ◽

pp. 65-68 ◽

Cited By ~ 4

Author(s):

Liaqat Ali Khan

Keyword(s):

Continuous Functions ◽

Weighted Spaces ◽

Regular Space ◽

Local Convexity ◽

Range Space ◽

Completely Regular ◽

Spaces Of Continuous Functions ◽

Vector Valued Functions ◽

Vector Valued ◽

Locally Convex

The fundamental work on approximation in weighted spaces of continuous functions on a completely regular space has been done mainly by Nachbin ([5], [6]). Further investigations have been made by Summers [10], Prolla ([7], [8]), and other authors (see the monograph [8] for more references). These authors considered functions with range contained in the scalar field or a locally convex topological vector space. In the present paper we prove some approximation results without local convexity of the range space.

Characterizations of the algebra of all real-valued continuous functions on a completely regular space

Illinois Journal of Mathematics ◽

10.1215/ijm/1255455003 ◽

1959 ◽

Vol 3 (1) ◽

pp. 121-133 ◽

Cited By ~ 8

Author(s):

Frank W. Anderson ◽

Robert L. Blair

Keyword(s):

Continuous Functions ◽

Regular Space ◽

Completely Regular

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

Bukovinian Mathematical Journal ◽

10.31861/bmj2021.01.21 ◽

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.

A Class of Function Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1960-029-3 ◽

1960 ◽

Vol 12 ◽

pp. 353-362 ◽

Cited By ~ 3

Author(s):

F. W. Anderson

Keyword(s):

Continuous Functions ◽

Regular Space ◽

Locally Compact ◽

Completely Regular ◽

Compact Spaces ◽

The Past ◽

Function Algebras ◽

Bounded Continuous Functions ◽

Locally Compact Spaces

A problem which has generated considerable interest during the past couple of decades is that of characterizing abstractly systems of realvalued continuous functions with various algebraic or topological-algebraic structures. With few exceptions known characterizations are of systems of bounded continuous functions on compact or locally compact spaces. Only recently have characterizations been given of the systems C(X) of all realvalued continuous functions on an arbitrary completely regular space X (1). One of the main objects of this paper is to provide, by using certain special techniques, a characterization of C(X) for a particular class of (not necessarily compact) completely regular spaces.

Structure of continuous functions of a completely regular space

Mathematical Notes ◽

10.1007/bf01149928 ◽

1976 ◽

Vol 19 (6) ◽

pp. 505-508

Author(s):

V. V. Pashenkov

Keyword(s):

Continuous Functions ◽

Regular Space ◽

Completely Regular

Joint Continuity of Separately Continuous Mappings with Values in Completely Regular Spaces

Tatra Mountains Mathematical Publications ◽

10.1515/tmmp-2017-0004 ◽

2017 ◽

Vol 68 (1) ◽

pp. 47-58 ◽

Cited By ~ 1

Author(s):

Volodymyr Maslyuchenko ◽

Oksana Myronyk ◽

Olha Filipchuk

Keyword(s):

Continuous Functions ◽

Regular Space ◽

Joint Continuity ◽

Completely Regular ◽

Several Variables ◽

Continuous Mappings

Abstract We prove general theorems on quasi-continuity of mappings f : X1 × ⋯ × Xn → Z with values in a completely regular space Z. As consequences, we obtain results on joint continuity of separately continuous functions of several variables involving the previous results of several authors.