scholarly journals Completion of lattices of semi-continuous functions

1978 ◽  
Vol 26 (4) ◽  
pp. 453-464 ◽  
Author(s):  
John August ◽  
Charles Byrne
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Distributive Lattices ◽  
Normal Completion ◽  
Bitopological Spaces

AbstractIf U and V are toplogies on an abstract set x, then the triple (X, U, V) is a bitopologica space. Using the theorem of Priestley on the representation of distributive lattices, results of Dilworth concerning the normal completion of the lattice of bounded, continuous, realvalued functions on a topological space are extended to include the lattice of bounded, semi-continuous, real-valued functions on certain bitopological spaces. The distributivity of certain lattices is investigated, and the theorem of Funayama on distributive normal completions is generalized.

scholarly journals Some Modifications of Pairwise Soft Sets and Some of Their Related Concepts

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1781
Author(s):  
Samer Al Ghour
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Soft Sets ◽  
New Concepts ◽  
Bitopological Spaces ◽  
The Relationship ◽  
Soft Topological Space

In this paper, we first define soft u-open sets and soft s-open as two new classes of soft sets on soft bitopological spaces. We show that the class of soft p-open sets lies strictly between these classes, and we give several sufficient conditions for the equivalence between soft p-open sets and each of the soft u-open sets and soft s-open sets, respectively. In addition to these, we introduce the soft u-ω-open, soft p-ω-open, and soft s-ω-open sets as three new classes of soft sets in soft bitopological spaces, which contain soft u-open sets, soft p-open sets, and soft s-open sets, respectively. Via soft u-open sets, we define two notions of Lindelöfeness in SBTSs. We discuss the relationship between these two notions, and we characterize them via other types of soft sets. We define several types of soft local countability in soft bitopological spaces. We discuss relationships between them, and via some of them, we give two results related to the discrete soft topological space. According to our new concepts, the study deals with the correspondence between soft bitopological spaces and their generated bitopological spaces.

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.

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 On certain analogues of linkedness and supercompactness

2020 ◽  
Vol 55 ◽  
pp. 113-134
Author(s):  
A.G. Chentsov
Keyword(s):  
Topological Space ◽  
The Family ◽  
Measurable Structure ◽  
Bitopological Spaces ◽  
Algebra Of Sets

Natural generalizations of properties of the family linkedness and the topological space supercompactness are considered. We keep in mind reinforced linkedness when nonemptyness of intersection of preassigned number of sets from a family is postulated. Analogously, supercompactness is modified: it is postulated the existence of an open subbasis for which, from every covering (by sets of the subbasis), it is possible to extract a subcovering with a given number of sets (more precisely, not more than a given number). It is clear that among all families having the reinforced linkedness, one can distinguish families that are maximal in ordering by inclusion. Under natural and (essentially) “minimal”' conditions imposed on the original measurable structure, among the mentioned maximal families with reinforced linkedness, ultrafilters are certainly contained. These ultrafilters form subspaces in the sense of natural topologies corresponding conceptually to schemes of Wallman and Stone. In addition, maximal families with reinforced linkedness, when applying topology of the Wallman type, have the above-mentioned property generalizing supercompactness. Thus, an analogue of the superextension of the $T_1$-space is realized. The comparability of “Wallman”' and “Stone”' topologies is established. As a result, bitopological spaces (BTS) are realized; for these BTS, under equipping with analogous topologies, ultrafilter sets are subspaces. It is indicated that some cases exist when the above-mentioned BTS is nondegenerate in the sense of the distinction for forming topologies. At that time, in the case of “usual” linkedness (this is a particular case of reinforced linkedness), very general classes of spaces are known for which the mentioned BTS are degenerate (the cases when origial set, i.e., “unit”' is equipped with an algebra of sets or a topology).

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.

scholarly journals A Lemma on Continuous Functions

1960 ◽  
Vol 3 (2) ◽  
pp. 186-187
Author(s):  
J. Lipman
Keyword(s):  
Topological Space ◽  
Fundamental Group ◽  
Closed Interval ◽  
Continuous Functions ◽  
The Other ◽  
Special Consideration ◽  
Boundary Lines

The point of this note is to get a lemma which is useful in treating homotopy between paths in a topological space [1].As explained in the reference, two paths joining a given pair of points in a space E are homotopic if there exists a mapping F: I x I →E (I being the closed interval [0,1] ) which deforms one path continuously into the other. In practice, when two paths are homotopic and the mapping F is constructed, then the verification of all its required properties, with the possible exception of continuity, is trivial. The snag occurs when F is a combination of two or three functions on different subsets of I x I. Then the boundary lines between these subsets have to be given special consideration, and although the problems resulting are routine their disposal can involve some tedious calculation and repetition. In the development [l] of the fundamental group of a space, for example, this sort of situation comes up four or five times.

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 Approximation and extension of continuous functions

1994 ◽  
Vol 57 (2) ◽  
pp. 149-157 ◽  
Author(s):  
Salvador Hernández-Muñoz
Keyword(s):  
Topological Space ◽  
Short Proof ◽  
Extension Theorem ◽  
Continuous Functions ◽  
Vector Valued

AbstractIn this paper we study the approximation of vector valued continuous functions defined on a topological space and we apply this study to different problems. Thus we give a new proof of Machado's Theorem. Also we get a short proof of a Theorem of Katětov and we prove a generalization of Tietze's Extension Theorem for vector-valued continuous functions, thereby solving a question left open by Blair.

scholarly journals On generalizations of C*-embedding for wallman rings

1978 ◽  
Vol 25 (2) ◽  
pp. 215-229 ◽  
Author(s):  
H. L. Bentley ◽  
B. J. Taylor
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Topological Properties ◽  
Zero Sets ◽  
Algebraic Properties

AbstractBiles (1970) has called a subring A of the ring C(X), of all real valued continuous functions on a topological space X, a Wallman ring on X whenever Z(A), the zero sets of functions belonging to A, forms a normal base on X in the sense of Frink (1964). Previously, we have related algebraic properties of a Wallman ring A to topological properties of the Wallman compactification w(Z(A)) of X determined by the normal base Z(A). Here we introduce two different generalizations of the concept of “a C*-embedded subset” and study relationships between these and topological (respectively, algebraic) properties of w(Z(A)) (respectively, A).

scholarly journals rc-continuous functions and functions with rc-strongly closed graph

2003 ◽  
Vol 2003 (72) ◽  
pp. 4547-4555
Author(s):  
Bassam Al-Nashef
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Continuous Functions ◽  
Closed Graph ◽  
Compact Spaces ◽  
Extremally Disconnected ◽  
The Family ◽  
Regular Closed

The family of regular closed subsets of a topological space is used to introduce two concepts concerning a functionffrom a spaceXto a spaceY. The first of them is the notion offbeing rc-continuous. One of the established results states that a spaceYis extremally disconnected if and only if each continuous function from a spaceXtoYis rc-continuous. The second concept studied is the notion of a functionfhaving an rc-strongly closed graph. Also one of the established results characterizes rc-compact spaces (≡S-closed spaces) in terms of functions that possess rc-strongly closed graph.

Sign in / Sign up

Export Citation Format

Share Document