scholarly journals Lebesgue Measurability of Separately Continuous Functions and Separability

2007 ◽  
Vol 2007 ◽  
pp. 1-4
Author(s):  
V. V. Mykhaylyuk
Keyword(s):  
Topological Space ◽  
Chain Condition ◽  
Continuous Functions ◽  
Negative Answer ◽  
Regular Space ◽  
Property A ◽  
Completely Regular ◽  
Open Set ◽  
Countable Chain Condition ◽  
Countable Chain

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.

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 Perfect Mappings and Spaces of Countable Type

1970 ◽  
Vol 22 (6) ◽  
pp. 1208-1210 ◽  
Author(s):  
J. E. Vaughan
Keyword(s):  
Topological Space ◽  
Affirmative Answer ◽  
Regular Space ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Countable Type ◽  
Open Set

In [1, p. 41, Theorem 3.10] Arhangel'skiï proved that the perfect image of a completely regular space of countable type is of countable type, and he asked [1, p. 60, problem 4] if a similar result held for regular or Hausdorff spaces. In this paper, it is proved that the perfect image of a space of countable type is of countable type, provided that the image is Hausdorff or regular. An affirmative answer to both of Arhangel'skiï's questions follows immediately from this. Arhangel'skiï made use of the Stone-Čech compactification in the proof of his result, but the proofs below are of a different nature.Let X be a topological space and let K ⊂ X. A collection of open sets is called a base at K provided that for every open set W ⊃ K there exists such that K ⊂ U ⊂ W. Clearly, we may assume that every member of contains K.

On Countability of Point-Finite Families of Sets

1979 ◽  
Vol 31 (4) ◽  
pp. 673-679 ◽  
Author(s):  
Heikki J. K. Junnila
Keyword(s):  
Topological Space ◽  
Finite Measure ◽  
Chain Condition ◽  
Finite Family ◽  
Topological Spaces ◽  
Countable Chain Condition ◽  
The Family ◽  
Measure Spaces ◽  
Countable Chain ◽  
Separable Topological Space

It is well known that in a separable topological space every point-finite family of open subsets is countable. In the following we are going to show that both in σ-finite measure-spaces and in topological spaces satisfying the countable chain condition, point-finite families consisting of “large” subsets are countable.Notation and terminology. Let A be a set. The family consisting of all (finite) subsets of A is denoted by . Let be a family of subsets of A. The sets and are denoted by and , respectively. We say that the family is point-finite (disjoint) if for each a ∈ A , the family has at most finitely many members (at most one member).

Stalking the Souslin Tree—A Topological Guide

1976 ◽  
Vol 19 (3) ◽  
pp. 337-341 ◽  
Author(s):  
Franklin D. Tall
Keyword(s):  
Topological Space ◽  
Chain Condition ◽  
Local Connectivity ◽  
First Category ◽  
Countable Chain Condition ◽  
Countable Chain ◽  
Dense Sets ◽  
Natural Way

It has long been known that the existence of a Souslin line entails (and is entailed by) the existence of a Souslin tree; indeed such a tree can be built from the open subsets of the line in a natural way. It will be shown that less onerous restrictions on a topological space than orderability allow the construction to proceed. For example, to the expected requirements-that the space satisfy the countable chain condition and not be separable, one can add the hypothesis of local connectivity, and that either first category sets be nowhere dense or that nowhere dense sets be separable.

scholarly journals A note on rings of continous functions

1978 ◽  
Vol 1 (1) ◽  
pp. 87-92
Author(s):  
J. S. Yang
Keyword(s):  
Topological Space ◽  
Large Class ◽  
Algebraic Structure ◽  
Continuous Functions ◽  
Regular Space ◽  
Topological Ring ◽  
Completely Regular ◽  
Pointwise Multiplication ◽  
Path Connected

For a topological spaceX, and a topological ringA, letC(X,A)be the ring of all continuous functions fromXintoAunder the pointwise multiplication. We show that the theorem “there is a completely regular spaceYassociated with a given topological spaceXsuch thatC(Y,R)is isomorphic toC(X,R)” may be extended to a fairly large class of topologlcal rings, and that, in the study of algebraic structure of the ringC(X,A), it is sufficient to studyC(X,R)ifAis path connected.

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

1975 ◽  
Vol 19 (3) ◽  
pp. 221-229 ◽  
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).

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