Mathematics

Given a continuum X and p∈X , we will consider the hyperspace C(p,X) of all subcontinua of X containing p . Given a family of continua C , a continuum X∈C and p∈X , we say that (X,p) has unique hyperspace C(p,X) relative to C if for each Y∈C and q∈Y such that C(p,X) and C(q,Y) are homeomorphic, then there is an homeomorphism between X and Y sending p to q . In this paper we show that (X,p) has unique hyperspace C(p,X) relative to the classes of dendrites if and only if X is a tree, we present also some classes of continua without unique hyperspace C(p,X) ; this answer some questions posed in \cite{this http URL(2019)}.

Convergence theory is an extension of general topology. In contrast with topology, it is closed under some important operations, like exponentiation. With all its advantages, convergence theory remains rather unknown. It is an aim of this paper to make it more familiar to the mathematical community.

With a complete Heyting algebra L as the truth value table, we prove that the collections of open filters of stratified L -valued topological spaces form a monad. By means of L -Scott topology and the specialization L -order, we get that the algebras of open filter monad are precisely L -continuous lattices.

In [A. Day, Filter monads, continuous lattices and closure systems, Can. J. Math. 27 (1975) 50--59], Day showed that continuous lattices are precisely the algebras of the open filter monad over the category of T 0 spaces. The aim of this paper is to give a clean and clear version of the whole process of Day's approach.

Assuming the existence of c incomparable selective ultrafilters, we classify the non-torsion Abelian groups of cardinality c that admit a countably compact group topology. We show that for each κ∈[c, 2 c ] each of these groups has a countably compact group topology of weight κ without non-trivial convergent sequences and another that has convergent sequences. Assuming the existence of 2 c selective ultrafilters, there are at least 2 c non homeomorphic such topologies in each case and we also show that every Abelian group of cardinality at most 2 c is algebraically countably compact. We also show that it is consistent that every Abelian group of cardinality c that admits a countably compact group topology admits a countably compact group topology without non-trivial convergent sequences whose weight has countable cofinality.

We investigate the Eilenberg-Moore algebras of the extended probabilistic powerdomain monad V w over the category TOP 0 of T 0 topological spaces and continuous maps. We prove that every V w -algebra in our setting is a weakly locally convex sober topological cone, and that a map is the structure map of a V w -algebra if and only if it is continuous and sends every continuous valuation to its unique barycentre. Conversely, for locally linear sober cones (a strong form of local convexity), the mere existence of barycentres entails that the barycentre map is the structure map of a V w -algebra; moreover the algebra morphisms are exactly the linear continuous maps in that case. We also examine the algebras of two related monads, the simple valuation monad V f and the point-continuous valuation monad V p . In TOP 0 their algebras are fully characterised as weakly locally convex topological cones and weakly locally convex sober topological cones, respectively. In both cases, the algebra morphisms are continuous linear maps between the corresponding algebras.

We have defined almost separable space. We show that like separability, almost separability is c productive and converse also true under some restrictions. We establish a Baire Category theorem like result in Hausdorff, Pseudocompacts spaces. We investigate few relationships among separability, almost separability, sequential separability, strongly sequential separability.

We explore almost-normality in Isbell-Mrówka spaces and some related concepts. We use forcing to provide an example of an almost-normal not normal almost disjoint family, explore the concept of semi-normality in Isbell-Mrówka spaces, define the concept of strongly ( ℵ 0 ,<c) -separated almost disjoint families and prove the generic existence of completely separable strongly ( ℵ 0 ,<c) -almost disjoint families assuming s=c and b=c . We also provide an example of a Tychonoff almost-normal not normal pseudocompact space which is not countably compact, answering a question from P. Szeptycki and S. Garcia-Balan.

In 1999, Molodtsov initiated the concept of Soft Sets Theory as a new mathematical tool and a completely different approach for dealing with uncertainties in many fields of applied sciences. In 2011, Shabir and Naz introduced and studied the theory of soft topological spaces, also defining and investigating many new soft properties as generalization of the classical ones. In this paper, we introduce the notions of soft separation between soft points and soft closed sets in order to obtain a generalization of the well-known Embedding Lemma for soft topological spaces.

We define a class of spaces on which one may generalise the notion of compactness following motivating examples from higher-dimensional number theory. We establish analogues of several well-known topological results (such as Tychonoff's Theorem) for such spaces. We also discuss several possible applications of this framework, including the theory of harmonic analysis on non-locally compact groups.