Complete Hausdorffness and Complete Regularity on Supra Topological Spaces

The supra topological topic is of great importance in preserving some topological properties under conditions weaker than topology and constructing a suitable framework to describe many real-life problems. Herein, we introduce the version of complete Hausdorffness and complete regularity on supra topological spaces and discuss their fundamental properties. We show the relationships between them with the help of examples. In general, we study them in terms of hereditary and topological properties and prove that they are closed under the finite product space. One of the issues we are interested in is showing the easiness and diversity of constructing examples that satisfy supra T i spaces compared with their counterparts on general topology.

Regularities in Ordered n-Ary Semihypergroups

This paper deals with a class of hyperstructures called ordered n-ary semihypergroups which are studied by means of j-hyperideals for all positive integers 1≤j≤n and n≥3. We first introduce the notion of (softly) left regularity, (softly) right regularity, (softly) intra-regularity, complete regularity, generalized regularity of ordered n-ary semihypergroups and investigate their related properties. Several characterizations of them in terms of j-hyperideals are provided. Finally, the relationships between various classes of regularities in ordered n-ary semihypergroups are also established.

Some New Variants of Relative Regularity via Regularly Closed Sets

Every topological property can be associated with its relative version in such a way that when smaller space coincides with larger space, then this relative property coincides with the absolute one. This notion of relative topological properties was introduced by Arhangel’skii and Ganedi in 1989. Singal and Arya introduced the concepts of almost regular spaces in 1969 and almost completely regular spaces in 1970. In this paper, we have studied various relative versions of almost regularity, complete regularity, and almost complete regularity. We investigated some of their properties and established relationships of these spaces with each other and with the existing relative properties.

The B-topology on S ∗-doubly quasicontinuous posets

The notions of o s {o}_{s} -convergence and S ∗ {S}^{\ast } -doubly quasicontinuous posets are introduced, which can be viewed as common generalizations of Birkhoff’s order-convergence and S ∗ {S}^{\ast } -doubly continuous posets, respectively. We first consider the relationship between o s {o}_{s} -convergence and B-topology and show that the topology induced by o s {o}_{s} -convergence according to the standard topological approach is the B-topology precisely. Then, the topological characterization for the S ∗ {S}^{\ast } -doubly quasicontinuity is presented. It is proved that a poset is S ∗ {S}^{\ast } -doubly quasicontinuous iff the poset equipped with the B-topology is locally hyperclosed iff the lattice of all B-open subsets of the poset is hypercontinuous. Finally, the order theoretical condition for the o s {o}_{s} -convergence being topological is given and the complete regularity of B-topology on S ∗ {S}^{\ast } -doubly quasicontinuous posets is explored.

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.

Complete regularity of Ellis semigroups of -actions

It is shown that the Ellis semigroup of a $\mathbb Z$ -action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a $\mathbb Z$ - or $\mathbb R$ -action for which forward proximality and backward proximality are transitive relations is shown to have at most two left minimal ideals. Finally, the notion of near simplicity of the Ellis semigroup is introduced and related to the above.

Regularity analysis for an abstract thermoelastic system with inertial term

In this paper, we provide a complete regularity analysis for an abstract system of coupled hyperbolic and parabolic equations. The asymptotic stability of this model was investigated in [6]. We are able to decompose the parameter region into three parts where the semigroup associated with the system is analytic, of Gevrey class of specific order, and non-smoothing, respectively. Moreover, by a detailed and creative spectral analysis,, we will show that the order of Gevrey class is sharp, under proper conditions. We also show that the orders of the polynomial stability obtained in [6] is optimal.

An internal characterization of complete regularity

This note presents a short proof of an internal characterization of complete regularity.

On Epicompletely Regularity

A topological space (X, τ) is said to be epicompletely regular if there is a coarser topological space (X, τ′) that is Tychonoff. In this study we investigated this natural type of complete regularity. We give some examples to explain some relationships between epicompletely regular space and some separation axioms.