Baire-Type Properties in Metrizable c0(Ω, X)

Ferrando and Lüdkovsky proved that for a non-empty set Ω and a normed space X, the normed space c0(Ω,X) is barrelled, ultrabornological, or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological, or unordered Baire-like. When X is a metrizable locally convex space, with an increasing sequence of semi-norms .n∈N defining its topology, then c0(Ω,X) is the metrizable locally convex space over the field K (of the real or complex numbers) of all functions f:Ω→X such that for each ε>0 and n∈N the set ω∈Ω:f(ω)n>ε is finite or empty, with the topology defined by the semi-norms fn=supf(ω)n:ω∈Ω, n∈N. Kąkol, López-Pellicer and Moll-López also proved that the metrizable space c0(Ω,X) is quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p if and only if X is, respectively, quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p. The main result of this paper is that the metrizable c0(Ω,X) is baireled if and only if X is baireled, and its proof is divided in several lemmas, with the aim of making it easier to read. An application of this result to closed graph theorem, and two open problems are also presented.

Reconfiguring Dominating Sets in Minor-Closed Graph Classes

For a graph G, two dominating sets D and $$D'$$ D ′ in G, and a non-negative integer k, the set D is said to k-transform to $$D'$$ D ′ if there is a sequence $$D_0,\ldots ,D_\ell $$ D 0 , … , D ℓ of dominating sets in G such that $$D=D_0$$ D = D 0 , $$D'=D_\ell $$ D ′ = D ℓ , $$|D_i|\le k$$ | D i | ≤ k for every $$i\in \{ 0,1,\ldots ,\ell \}$$ i ∈ { 0 , 1 , … , ℓ } , and $$D_i$$ D i arises from $$D_{i-1}$$ D i - 1 by adding or removing one vertex for every $$i\in \{ 1,\ldots ,\ell \}$$ i ∈ { 1 , … , ℓ } . We prove that there is some positive constant c and there are toroidal graphs G of arbitrarily large order n, and two minimum dominating sets D and $$D'$$ D ′ in G such that Dk-transforms to $$D'$$ D ′ only if $$k\ge \max \{ |D|,|D'|\}+c\sqrt{n}$$ k ≥ max { | D | , | D ′ | } + c n . Conversely, for every hereditary class $$\mathcal{G}$$ G that has balanced separators of order $$n\mapsto n^\alpha $$ n ↦ n α for some $$\alpha <1$$ α < 1 , we prove that there is some positive constant C such that, if G is a graph in $$\mathcal{G}$$ G of order n, and D and $$D'$$ D ′ are two dominating sets in G, then Dk-transforms to $$D'$$ D ′ for $$k=\max \{ |D|,|D'|\}+\lfloor Cn^\alpha \rfloor $$ k = max { | D | , | D ′ | } + ⌊ C n α ⌋ .

The Reverse of the Intermediate Value Theorem in Some Topological Spaces

Any continuous function with values in a Hausdorff topological space has a closed graph and satisfies the property of intermediate value. However, the reverse implications are false, in general. In this article, we treat additional conditions on the function, and its graph for the reverse to be true.

Applications of some operators on supra topological spaces

In this paper, the notion of an operator \gamma on a supra topological space (X,\mu ) is studied and then utilized to analyze supra \gamma -open sets. The notions of {\mu }_{\gamma }-g.closed sets on the subspace are introduced and investigated. Furthermore, some new {\mu }_{\gamma }-separation axioms are formulated and the relationships between them are shown. Moreover, some characterizations of the new functions via operator \gamma on \mu are presented and investigated. Finally, we give some properties of {S}_{(\gamma ,\beta )}-closed graph and strongly {S}_{(\gamma ,\beta )}-closed graph.

Points of Continuity of Quasi-Continuous Functions

We study points of joint continuity of multi-variable separately quasi-continuous functions which are continuous with respect to one variable. We will show that the assumption of complete regularity in the paper of V. Maslyuchenko et al in ”Tatra Mt. Math. Pub. 68, (2017), 47–58” can be removed in some situations. Moreover, we use a topological game argument to prove that the set of points of continuity of functions with closed graph into ωΔ-spaces is a Gδ subset of its domain provided that its domain is a W-space.

Fuzzy Open Mapping and Fuzzy Closed Graph Theorems in Fuzzy Length Space

The theory of fuzzy set includes many aspects that regard important and significant in different fields of science and engineering in addition to there applications. Fuzzy metric and fuzzy normed spaces are essential structures in the fuzzy set theory. The concept of fuzzy length space has been given analogously and the properties of this space are studied few years ago. In this work, the definition of a fuzzy open linear operator is presented for the first time and the fuzzy Barise theorem is established to prove the fuzzy open mapping theorem in a fuzzy length space. Finally, the definition of a fuzzy closed linear operator on fuzzy length space is introduced to prove the fuzzy closed graph theorem.