Pythagorean Fuzzy Rough Connected Spaces

In this paper Pythagorean fuzzy rough set and Pythagorean fuzzy rough topological spaces are defined for the connected space. Then, the properties of connectedness are discussed with examples.

Sum connectedness in proximity spaces

The notion of sum δ-connected proximity spaces which contain the category of δ-connected and locally δ-connected spaces is defined. Several characterizations of it are substantiated. Weaker forms of sum δ-connectedness are also studied.

Infra -α- Compact and Infra -α- Connected Spaces

In 2016 Hakeem A. Othman and Md. Hanif Page introduced a new notion of set in general topology called an infra -α- open set and investigated its fundamental properties and studied the relationship between infra -α- open set and other topological sets. The objective of this paper is to introduce the new concepts called infra -α- compact space, countably infra -α- compact space, infra -α- Lindelof space, almost infra -α- compact space, mildly infra -α- compact space and infra -α- connected space in general topology and investigate several properties and characterizations of these new concepts in topological spaces.

Connectedness and Local Connectedness on Infra Soft Topological Spaces

This year, we introduced the concept of infra soft topology as a new generalization of soft topology. To complete our analysis of this space, we devote this paper to presenting the concepts of infra soft connected and infra soft locally connected spaces. We provide some descriptions for infra soft connectedness and elucidate that there is no relationship between an infra soft topological space and its parametric infra topological spaces with respect to the property of infra soft connectedness. We discuss the behaviors of infra soft connected and infra soft locally connected spaces under infra soft homeomorphism maps and a finite product of soft spaces. We complete this manuscript by defining a component of a soft point and establishing its main properties. We determine the conditions under which the number of components is finite or countable, and we discuss under what conditions the infra soft connected subsets are components.

Goodwillie Approximations to Higher Categories

We construct a Goodwillie tower of categories which interpolates between the category of pointed spaces and the category of spectra. This tower of categories refines the Goodwillie tower of the identity functor in a precise sense. More generally, we construct such a tower for a large class of ∞ \infty -categories C \mathcal {C} and classify such Goodwillie towers in terms of the derivatives of the identity functor of C \mathcal {C} . As a particular application we show how this provides a model for the homotopy theory of simply-connected spaces in terms of coalgebras in spectra with Tate diagonals. Our classification of Goodwillie towers simplifies considerably in settings where the Tate cohomology of the symmetric groups vanishes. As an example we apply our methods to rational homotopy theory. Another application identifies the homotopy theory of p p -local spaces with homotopy groups in a certain finite range with the homotopy theory of certain algebras over Ching’s spectral version of the Lie operad. This is a close analogue of Quillen’s results on rational homotopy.

Exploring ‘digital placemaking’

This brief essay introduces the special issue on the topic of ‘digital placemaking’ – a concept describing the use of digital media to create a sense of place for oneself and/or others. As a broad framework that encompasses a variety of practices used to create emotional attachments to place through digital media use, digital placemaking can be examined across a variety of domains. The concept acknowledges that, at its core, a drive to create and control a sense of place is understood as primary to how social actors identify with each other and express their identities and how communities organize to build more meaningful and connected spaces. This idea runs through the articles in the issue, exploring the many ways people use digital media, under varied conditions, to negotiate differential mobilities and become placemakers – practices that may expose or amplify preexisting inequities, exclusions, or erasures in the ways that certain populations experience digital media in place and placemaking.

New types of locally connected spaces via clopen set

In this paper, we define and study a new type of connected spaces called λco-connected space. It is remarkable that the class of λ-connected spaces is a subclass of the class of λco-connected spaces. We discuss some characterizations and properties of λco-connected spaces, λco components and λco-locally connected spaces

On perfectly αδ-continuous functions in topological spaces

The concept of αδ-closed sets was introduced in the research paper “On Strongly-αδ-Super-Irresolute Functions In Topological Spaces. The aim of this paper is to consider and characterize αδ-irresolute and αδ-continuous functions via the concept of αδ-closed sets and to relate these concepts to the classes of αδ-compact spaces and αδ0-connected spaces.