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More on μ-semi-Lindelöf sets in μ-spaces

Sarsak Mohammad S.

Demonstratio Mathematica

2021

Vol. 54, nr 1

259--271

Sarsak [On μ-compact sets in μ-spaces, Questions Answers Gen. Topology 31 (2013), no. 1, 49-57] introduced and studied the class of μ-Lindelöf sets in μ-spaces. Mustafa [μ-semi compactness and μ-semi Lindelöfness in generalized topological spaces, Int. J. Pure Appl. Math. 78 (2012), no. 4, 535-541] introduced and studied the class of μ-semi-Lindelöf sets in generalized topological spaces (GTSs); the primary purpose of this paper is to investigate more properties and mapping properties of μ-semi-Lindelöf sets in μ-spaces. The class of μ-semi-Lindelöf sets in μ-spaces is a proper subclass of the class of μ-Lindelöf sets in μ-spaces. It is shown that the property of being μ-semi-Lindelöf is not monotonic, that is, if (X, μ) is a μ-space and A ⊂ B ⊂ X, where A is μB-semi-Lindelöf, then A need not be μ-semi-Lindelöf. We also introduce and study a new type of generalized open sets in GTSs, called ωμ-semi-open sets, and investigate them to obtain new properties and characterizations of μ-semi-Lindelöf sets in μ-spaces.

The unified theory of certain types of generalizations of Lindelof spaces

Noiri T., Popa V.

Demonstratio Mathematica

2010

Vol. 43, nr 1

203-212

We introduce a new class of sets called u-m-open sets which are defined on a family of sets satisfying m-structures with the property of being closed under arbitrary union. The sets enable us to obtain some unified properties of certain types of generalizations of Lindelof spaces.

On the compactness and countable compactness of 2R in ZF

Keremedis K., Felouzis E., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 4

293-302

In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements "2R is countably compact" and "2R is compact".

Remark on the abstract Dirichlet problem for Baire-one functions

Kalenda O. F. K.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 1

55-73

We study the possibility of extending any bounded Baire-one function on the set of extreme points of a compact convex set to an affine Baire-one function and related questions. We give complete solutions to these questions within a class of Choquet simplices introduced by P. J. Stacey (1979). In particular we get an example of a Choquet simplex such that its set of extreme points is not Borel but any bounded Baire-one function on the set of extreme points can be extended to an affine Baire-one function. We also study the analogous questions for functions of higher Baire classes.