We show that the extent of a Tychonoff space of countable weak extent can be arbitrary big. The extent of X is e(X) = sup{|F | : F ⊂ X is closed and discrete} while we(X) = min{τ : for every open cover U of X there is A ⊂ X such that |A| ≤ τ and St(A,U) = X} is the weak extent of X (also called the star-Lindelöf number of X). Also we show that the extent of a normal space with countable weak ex...

In this talk we introduce and study quaternion-valued positive definite functions on locally compact Abelian groups, real countably Hilbertian nuclear spaces and on the space of countably infinite tuples of real numbers endowed with the Tychonoff topology. In particular, we prove a quaternionic version of the Bochner-Minlos theorem. We will see that in all these various settings the integral re...

There is a natural bijective correspondence between the compactifications of a Tychonoff space X, the totally bounded uniformities on X, and the unital C∗-subalgebras of C∗(X) (the algebra of bounded continuous complex valued functions on X) with what we call the completely regular separation property. The correspondence of compactifications with totally bounded uniformities is well know and ca...

It is known that the free topological group over the Tychonoff space X, denoted F(X), is a P space if and only if X is a P -space. This article is concerned with the question of whether one can characterize when F(X) is a weak P -space, that is, a space where all countable subsets are closed. Our main result is that F(X) is a weak P -space if and only if X is a weak P -space and every countable...

In this paper we characterize various sorts of boundedness of the free (abelian) topological group F (X) (A(X)) as well as the free locally-convex linear topological space L(X) in terms of properties of a Tychonoff space X. These properties appear to be close to so-called selection principles, which permits us to show, that (it is consistent with ZFC that) the property of Hurewicz (Menger) is l...

We connect the dual adjunction between MV-algebras and Tychonoff spaces with the general theory of natural dualities, and provide a number of applications. In doing so, we simplify the aforementioned construction by observing that there is no need of using presentations of MV-algebras in order to obtain the adjunction. We also provide a description of the dual maps that is intrinsically geometr...

We show that the unit ball of lp(Γ) in its weak topology is a continuous image of σ1(Γ) and we deduce some combinatorial properties of its lattice of open sets which are not shared by the balls of other equivalent norms when Γ is uncountable. For a set Γ and a real number 1 < p < ∞, the Banach space lp(Γ) is a reflexive space, hence its unit ball is compact in the weak topology and in fact, it ...

For a Tychonoff space X, the lattice UX was introduced in L. A. Pérez-Morales, G. Delgadillo-Piñón, and R. Pichardo-Mendoza, The of uniform topologies on C(X), Questions Answers General Topology 39 (2021), 65-71. In present paper we continue study UX. To be specific, deals, its first half, with structural categorical properties UX, while second part focuses cardinal characteristics how these re...

The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in number ways, including as colimit homeomorphism groups with given compact support, or subgroup the its Stone-\v{C}ech compactification. A is said to have Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. authors provide necessary and sufficient conditions for finite produ...

We call a topological space X locally compact with defects if all points in possess neighborhoods except for some points. investigate this weaker version of local compactness. show that x ∈ X• the partition singletons X\(X• ∪ (U\U)) is finite, where U ̸= an open neighborhood x, then Tychonoff space. Let be T1c such each has union pairwise disjoint subsets S s∈S Fs. Then, we family {Fs}s∈S finite...