Quasihomeomorphisms and Skula spaces

2021 ◽  
Author(s):  
Othman Echi
Keyword(s):  
Topological Space ◽  
Alternative Proof ◽  
Manuscripta Math ◽  
Alexandroff Space ◽  
Closed Sets ◽  
Locally Closed Sets ◽  
Reflection Formula

Let [Formula: see text] be a topological space. By the Skula topology (or the [Formula: see text]-topology) on [Formula: see text], we mean the topology [Formula: see text] on [Formula: see text] with basis the collection of all [Formula: see text]-locally closed sets of [Formula: see text], the resulting space [Formula: see text] will be denoted by [Formula: see text]. We show that the following results hold: (1) [Formula: see text] is an Alexandroff space if and only if the [Formula: see text]-reflection [Formula: see text] of [Formula: see text] is a [Formula: see text]-space. (2) [Formula: see text] is a Noetherian space if and only if [Formula: see text] is finite. (3) If we denote by [Formula: see text] the Alexandroff extension of [Formula: see text], then [Formula: see text] if and only if [Formula: see text] is a Noetherian quasisober space. We also give an alternative proof of a result due to Simmons concerning the iterated Skula spaces, namely, [Formula: see text]. A space is said to be clopen if its open sets are also closed. In [R. E. Hoffmann, Irreducible filters and sober spaces, Manuscripta Math. 22 (1977) 365–380], Hoffmann introduced a refinement clopen topology [Formula: see text] of [Formula: see text]: The indiscrete components of [Formula: see text] are of the form [Formula: see text], where [Formula: see text] and [Formula: see text] is the intersection of all open sets of [Formula: see text] containing [Formula: see text] (equivalently, [Formula: see text]). We show that [Formula: see text]

scholarly journals Relations of Pre Generalized Regular Weakly Locally Closed Sets in Topological Spaces

2021 ◽  
Vol 12 (3) ◽  
pp. 3444-3454
Author(s):  
Vijayakumari T Et.al
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Closed Set ◽  
Closed Sets ◽  
Open Set ◽  
Continuous Maps ◽  
Locally Closed Sets

In this paper pgrw-locally closed set, pgrw-locally closed*-set and pgrw-locally closed**-set are introduced. A subset A of a topological space (X,t) is called pgrw-locally closed (pgrw-lc) if A=GÇF where G is a pgrw-open set and F is a pgrw-closed set in (X,t). A subset A of a topological space (X,t) is a pgrw-lc* set if there exist a pgrw-open set G and a closed set F in X such that A= GÇF. A subset A of a topological space (X,t) is a pgrw-lc**-set if there exists an open set G and a pgrw-closed set F such that A=GÇF. The results regarding pgrw-locally closed sets, pgrw-locally closed* sets, pgrw-locally closed** sets, pgrw-lc-continuous maps and pgrw-lc-irresolute maps and some of the properties of these sets and their relation with other lc-sets are established.

HOMOGENEOUS FUNCTIONALLY ALEXANDROFF SPACES

2017 ◽  
Vol 97 (2) ◽  
pp. 331-339 ◽  
Author(s):  
SAMI LAZAAR ◽  
TOM RICHMOND ◽  
HOUSSEM SABRI
Keyword(s):  
Topological Space ◽  
Alexandroff Space ◽  
Closed Sets

A function $f:X\rightarrow X$ determines a topology $P(f)$ on $X$ by taking the closed sets to be those sets $A\subseteq X$ with $f(A)\subseteq A$. The topological space $(X,P(f))$ is called a functionally Alexandroff space. We completely characterise the homogeneous functionally Alexandroff spaces.

scholarly journals Study of RL-Connectedness and RL-Compactness

2020 ◽  
pp. 117-123
Author(s):  
J. M. U. D. Wijerathne ◽  
P. Elango
Keyword(s):  
Topological Space ◽  
Closed Sets ◽  
Continuous Maps ◽  
Locally Closed Sets

In this paper, we introduce a new kind of locally closed sets called regular locally closed sets (brie y RL-closed sets) in a topological space which are weaker than the locally closed sets. Regular locally continuous maps and regular locally irresolute maps are also introduced and studied some of their properties. Finally, we introduce the concept of regular locally connectedness and regular locally compactness on a topological space using the RL-closed sets.

scholarly journals New notions in ideal topological space

2019 ◽  
Vol 38 (5) ◽  
pp. 19-31
Author(s):  
Chinnapazham Santhini ◽  
M. Suganya
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Closed Sets ◽  
New Class ◽  
Locally Closed Sets

In this paper, we apply the notion of I∗∗α g -closed sets to present and study a new class of locally closed sets called I∗∗α g -locally closed sets in ideal topological spaces along with their several characterizations and mutual relationships between the new notion and other locally closed sets. Further we introduce I∗∗α g-submaximal space and some properties of such notion are investigated.

scholarly journals Locally closed sets andLC-continuous functions

1989 ◽  
Vol 12 (3) ◽  
pp. 417-424 ◽  
Author(s):  
M. Ganster ◽  
I. L. Reilly
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Closed Set ◽  
Closed Sets ◽  
Generalized Continuity ◽  
Locally Closed Sets ◽  
Nearly Continuous

In this paper we introduce and study three different notions of generalized continuity, namely LC-irresoluteness, LC-continuity and sub-LC-continuity. All three notions are defined by using the concept of a locally closed set. A subset S of a topological space X is locally closed if it is the intersection of an open and a closed set. We discuss some properties of these functions and show that a function between topological spaces is continuous if and only if it is sub-LC-continuous and nearly continuous in the sense of Ptak. Several examples are provided to illustrate the behavior of these new classes of functions.

scholarly journals Contra-continuous functions and stronglyS-closed spaces

1996 ◽  
Vol 19 (2) ◽  
pp. 303-310 ◽  
Author(s):  
J. Dontchev
Keyword(s):  
Dense Subset ◽  
Continuous Functions ◽  
Closed Space ◽  
Closed Sets ◽  
Open Set ◽  
Locally Closed Sets ◽  
Continuous Images

In 1989 Ganster and Reilly [6] introduced and studied the notion ofLC-continuous functions via the concept of locally closed sets. In this paper we consider a stronger form ofLC-continuity called contra-continuity. We call a functionf:(X,τ)→(Y,σ)contra-continuous if the preimage of every open set is closed. A space(X,τ)is called stronglyS-closed if it has a finite dense subset or equivalently if every cover of(X,τ)by closed sets has a finite subcover. We prove that contra-continuous images of stronglyS-closed spaces are compact as well as that contra-continuous,β-continuous images ofS-closed spaces are also compact. We show that every stronglyS-closed space satisfies FCC and hence is nearly compact.

scholarly journals On generalized γµ-closed sets and related continuity

2021 ◽  
Vol 13 (2) ◽  
pp. 483-493
Author(s):  
Ritu Sen
Keyword(s):  
Topological Space ◽  
Main Interest ◽  
Closed Sets ◽  
Separation Axioms ◽  
Preservation Theorems ◽  
Generalized Topological Space ◽  
New Type ◽  
Weak Separation

Abstract In this paper our main interest is to introduce a new type of generalized open sets defined in terms of an operation on a generalized topological space. We have studied some properties of this newly defined sets. As an application, we have introduced some weak separation axioms and discussed some of their properties. Finally, we have studied some preservation theorems in terms of some irresolute functions.

Neutrosophic Nano Ideal Topological Structures

2018 ◽  
Author(s):  
Parimala Mani ◽  
Karthika M ◽  
jafari S ◽  
Smarandache F ◽  
Ramalingam Udhayakumar
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Topological Structures ◽  
Basic Properties ◽  
Closed Sets ◽  
Structural Space ◽  
New Type

Neutrosophic nano topology and Nano ideal topological spaces induced the authors to propose this new concept. The aim of this paper is to introduce a new type of structural space called neutrosophic nano ideal topological spaces and investigate the relation between neutrosophic nano topological space and neutrosophic nano ideal topological spaces. We define some closed sets in these spaces to establish their relationships. Basic properties and characterizations related to these sets are given.

Sign in / Sign up

Export Citation Format

Share Document