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The famous Ascoli-Arzelà theorem served as a springboard for research into compactness in function spaces, particularly spaces of continuous functions. This paper investigates compact subsets of spaces of continuous functions endowed with topologies between the topology of pointwise convergence and the topology of uniform convergence. More precisely, this paper studies necessary and sufficient conditions for a subset to be compact in Cλ,u(X) for a locally-λ space X when λ ⊇ 퓕(X), for a hemi- λ λf-space X when λ ⊆ 퓟 퓢(X), and for a k-space X when λ ⊇ 퓚(X). This paper also studies that every bounded subset of Cλ,u(X) has compact closure for some classes of topological spaces X. [ABSTRACT FROM AUTHOR]

In this paper, we define some combinatorial principles to characterize spaces X whose hyperspace satisfies some variation of some classical star selection principle. Specifically, the variations characterized are the selective and absolute versions of the star selection principles for the Menger and Rothberger cases; also, the hyperspaces considered in these characterizations are CL(X), 핂(X), 픽(X) and ℂ핊(X) in both cases, endowed with either the Fell topology or the Vietoris topology. [ABSTRACT FROM AUTHOR]

A space X is said to have the set-star-K-Menger property if for each nonempty subset A of X and for each sequence (퓤n : n ∈ ℕ) of collections of open sets in X such that for each n ∈ ℕ, A ⊆ ⋃ 퓤n, there is a sequence (Kn : n ∈ ℕ) of compact subsets of X such that A ⊆ ⋃ n ∈ N St(Kn, 퓤n). In this paper, we prove that: There exists a T1 set-star-Menger space which is not set-star-K-Menger and there exists a Tychonoff set-star-K-Menger space that is not set-star-Menger. Assuming 픡 = 픠, there exists a Tychonoff set-star-K-Menger space having a regular-closed Gδ-subspace which is not set-star-K-Menger. If the Alexandroff duplicate of a space X is set-star-K-Menger, then X is set-star-K-Menger. The product of set-star-K-Menger space and a compact space is rectangular set-star-K-Menger space. The above-mentioned results answer to Problem 2.5 and Problem 3.6, and give a partial answer to Problem 3.11 in [SINGH, S.: On set-star-K-Menger spaces, Publ. Math. Debrecen 100 (2022), 87–100]. Further, we continue to study the topological properties of set-star-K-Menger spaces. [ABSTRACT FROM AUTHOR]

Recently, the authors interested some new problems on multiplier spaces of Lorentz' almost convergence and fλ-convergence as a generalization of almost convergence. fλ-convergence is firstly introduced by Karakuş and Başar, and used for some new characterizations of completeness and barrelledness of the spaces through weakly unconditionally Cauchy series in a normed space X and its continuous dual X*. In the present paper, we deal with fλ-convergence to have some inclusion relations between the vector valued spaces obtained from this type convergence and corresponding classical sequence spaces, and to give new characterizations of some classical properties like completeness, reflexivity, Schur property and Grothendieck property of normed spaces. By the way, we give a characterization of finite-dimensional normed spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study some properties of homotopical closeness for paths. We define the quasi-small loop group as the subgroup of all classes of loops that are homotopically close to null-homotopic loops, denoted by π 1 q s (X , x) for a pointed space (X, x). Then we prove that, unlike the small loop group, the quasi-small loop group π 1 q s (X , x) does not depend on the base point, and that it is a normal subgroup containing π 1 s g (X , x) , the small generated subgroup of the fundamental group. Also, we show that a space X is homotopically path Hausdorff if and only if π 1 q s (X , x) is trivial. Finally, as consequences, we give some relationships between the quasi-small loop group and the quasi-topological fundamental group. [ABSTRACT FROM AUTHOR]

For a zero-dimensional Hausdorff space X, denote, as usual, by C(X, ℤ) the ring of continuous integer-valued functions on X. If f ∈ C(X, ℤ), denote by Z(f) the set of all points of X that are mapped to 0 by f. The set C K (X , Z) = { f ∈ C (X , Z) ∣ cl X (X ∖ Z (f)) is compact } is the integer-valued analogue of the ideal of functions with compact support in C(X). By first working in the category of locales and then interpreting the results in spaces, we characterize this ideal in several ways. Writing ζX for the Banaschewski compactification of X, we also explore some properties of ideals of C(X, ℤ) associated with subspaces of ζX analogously to how one associates, for any Tychonoff space Y, subsets of βY with ideals of C(Y). [ABSTRACT FROM AUTHOR]

Let F denote the factorable matrix and X ∈ {ℓp, c0, c, ℓ∞}. In this study, we introduce the domains X(F) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes (ℓp(F), ℓ∞), (ℓp(F), f) and (X, Y(F)) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F. Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix. [ABSTRACT FROM AUTHOR]