scholarly journals Characterizations of some near-continuous functions and near-open functions

1986 ◽  
Vol 9 (4) ◽  
pp. 715-720 ◽  
Author(s):  
C. W. Baker
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Open Set ◽  
Weakly Continuous ◽  
Almost Open ◽  
Open Maps

A subsetNof a topological space is defined to be aθ-neighborhood ofxif there exists an open setUsuch thatx∈U⫅C1   U⫅N. This concept is used to characterize the following types of functions: weakly continuous,θ-continuous, stronglyθ-continuous, almost stronglyθ-continuous, weaklyδ-continuous, weakly open and almost open functions. Additional characterizations are given for weaklyδ-continuous functions. The concept ofθ-neighborhood is also used to define the following types of open maps:θ-open, stronglyθ-open, almost stronglyθ-open, and weaklyδ-open functions.

scholarly journals Lebesgue Measurability of Separately Continuous Functions and Separability

2007 ◽  
Vol 2007 ◽  
pp. 1-4
Author(s):  
V. V. Mykhaylyuk
Keyword(s):  
Topological Space ◽  
Chain Condition ◽  
Continuous Functions ◽  
Negative Answer ◽  
Regular Space ◽  
Property A ◽  
Completely Regular ◽  
Open Set ◽  
Countable Chain Condition ◽  
Countable Chain

A connection between the separability and the countable chain condition of spaces withL-property (a topological spaceXhasL-property if for every topological spaceY, separately continuous functionf:X×Y→ℝand open setI⊆ℝ,the setf−1(I)is anFσ-set) is studied. We show that every completely regular Baire space with theL-property and the countable chain condition is separable and constructs a nonseparable completely regular space with theL-property and the countable chain condition. This gives a negative answer to a question of M. Burke.

scholarly journals On rings of real valued clopen continuous functions

2018 ◽  
Vol 19 (2) ◽  
pp. 203
Author(s):  
Susan Afrooz ◽  
Fariborz Azarpanah ◽  
Masoomeh Etebar
Keyword(s):  
Topological Space ◽  
Maximal Ideal ◽  
Quotient Space ◽  
Inverse Image ◽  
Continuous Functions ◽  
Discrete Space ◽  
Strong Continuity ◽  
Open Set ◽  
Maximal Ideals ◽  
Essential Ideal

<p>Among variant kinds of strong continuity in the literature, the clopen continuity or cl-supercontinuity (i.e., inverse image of every open set is a union of clopen sets) is considered in this paper.  We investigate and study the ring C<sub>s</sub>(X) of all real valued clopen continuous functions on a topological space X.  It is shown that every ƒ ∈ C<sub>s</sub>(X) is constant on each quasi-component in X and using this fact we show that C<sub>s</sub>(X) ≅ C(Y), where Y is a zero-dimensional s-quotient space of X.  Whenever X is locally connected, we observe  that C<sub>s</sub>(X) ≅ C(Y),  where Y is a discrete space.  Maximal ideals of C<sub>s</sub>(X) are characterized in terms of quasi-components in X and it turns out that X  is mildly compact(every clopen cover has a finite subcover) if and only if every maximal ideal  of C<sub>s</sub>(X)is  fixed. It is shown that the socle of C<sub>s</sub>(X) is  an essential ideal if and only if the union of all open quasi-components in X is s-dense.  Finally the counterparts of some familiar spaces, such as P<sub>s</sub>-spaces, almost P<sub>s</sub>-spaces, s-basically and s-extremally disconnected spaces  are  defined  and  some  algebraic  characterizations  of  them  are given via the ring C<sub>s</sub>(X).</p>

Real Functions, Covers and Bornologies

2021 ◽  
Vol 78 (1) ◽  
pp. 199-214
Author(s):  
Lev Bukovský
Keyword(s):  
Topological Space ◽  
Pointwise Convergence ◽  
Continuous Functions ◽  
Upper Semicontinuous ◽  
Upper Semicontinuous Functions ◽  
Covering Properties ◽  
Real Functions ◽  
Topology Of Pointwise Convergence ◽  
Semicontinuous Functions

Abstract The paper tries to survey the recent results about relationships between covering properties of a topological space X and the space USC(X) of upper semicontinuous functions on X with the topology of pointwise convergence. Dealing with properties of continuous functions C(X), we need shrinkable covers. The results are extended for A-measurable and upper A-semimeasurable functions where A is a family of subsets of X. Similar results for covers respecting a bornology and spaces USC(X) or C(X) endowed by a topology defined by using the bornology are presented. Some of them seem to be new.

scholarly journals A note on weakly θ-continuous functions

1989 ◽  
Vol 12 (1) ◽  
pp. 9-13 ◽  
Author(s):  
M. Mrševic ◽  
I. L. Reilly
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Topological Spaces ◽  
New Class ◽  
Weakly Continuous ◽  
Class Of Functions

Recently a new class of functions between topological spaces, called weaklyθ-continuous functions, has been introduced and studied. In this paper we show how an appropriate change of topology on the domain of a weaklyθ-continuous function reduces it to a weakly continuous function. This paper examines some of the consequences of this result.

Order continuous measures

1964 ◽  
Vol 60 (2) ◽  
pp. 205-207 ◽  
Author(s):  
G. T. Roberts
Keyword(s):  
Topological Space ◽  
Linear Form ◽  
Vector Lattice ◽  
Measure Zero ◽  
Continuous Functions ◽  
Order Convergence ◽  
Locally Compact ◽  
Compact Topological Space ◽  
Continuous Measures ◽  
Order Continuous

1. Objective. It is possible to define order convergence on the vector lattice of all continuous functions of compact support on a locally compact topological space. Every measure is a linear form on this vector lattice. The object of this paper is to prove that a measure is such that every set of the first category of Baire has measure zero if and only if the measure is a linear form which is continuous in the order convergence.

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 s-g-cocompact open set and Continuity

2020 ◽  
Vol 25 (2) ◽  
pp. 67-77 ◽  
Author(s):  
Raad Al-Abdulla ◽  
Salam Jabar
Keyword(s):  
Topological Space ◽  
Open Set ◽  
The Relationship

    Throughout this paper by a space we mean a supra topological space, we have studied some of propertiese to new set is called supra generalize- cocompact open set ( -g-coc-open set)and find the relation with other sets and our concluded anew class of the function called -g-coc-continuous, -g-coc'-continuous, -coc-continuous, -coc'-continuous We shall provided some properties of these concepts and it will explain the relationship among them and some results on this subjects are proved Throughout this work , and new concept have been illustrated including , -coc-ompact space .

scholarly journals On B-Open Sets

2019 ◽  
Vol 12 (2) ◽  
pp. 358-369
Author(s):  
Layth Muhsin Habeeb Alabdulsada
Keyword(s):  
Topological Space ◽  
Open Set

The aim of this paper is to introduce and study $\mathcal{B}$-open sets and related properties. Also, we define a bi-operator topological space $(X, \tau, T_1, T_2)$, involving the two operators $T_1$ and $T_2$, which are used to define $\mathcal{B}$-open sets. A $\mathcal{B}$-open set is, roughly speaking, a generalization of a $b$-open set, which is, in turn, a generalization of a pre-open set and a semi-open set. We introduce a number of concepts based on $\mathcal{B}$-open sets.

ON ALMOST WEAKLY CONTINUOUS FUNCTIONS

1998 ◽  
Vol 31 (2) ◽  
Author(s):  
Saeid Jafari ◽  
Takashi Noiri
Keyword(s):  
Continuous Functions ◽  
Weakly Continuous

Restrictive Semigroups of Continuous Functions on 0-Dimensional Spaces

1972 ◽  
Vol 24 (4) ◽  
pp. 598-611 ◽  
Author(s):  
Robert D. Hofer
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Analogous Problem

Let X be a topological space and Y a nonempty subspace of X. Γ(X, Y) denotes the semigroup under composition of all closed self maps of X which carry Y into Y, and is referred to as a restrictive semigroup of closed functions. Similarly, S(X, Y) is the analogous semigroup of continuous selfmaps of X, and is referred to as a restrictive semigroup of continuous functions. It is immediate that each homeomorphism from X onto U which carries the subspace Y of X onto the subspace V of U induces an isomorphism between Γ(X, Y) and Γ(U, V), and also an isomorphism between S(X, Y) and S(U, V). Indeed, one need only map f onto h o f o h-1. An isomorphism of this form is called representable. In [5, Theorem (3.1), p. 1223] it was shown that in most cases, each isomorphism from Γ(X, Y) onto Γ(U, V) is representable. The analogous problem was discussed for the semigroup S(X, Y) and it was pointed out by means of an example that one could not hope to obtain the same result for these semigroups without some further restrictions.

Sign in / Sign up

Export Citation Format

Share Document