scholarly journals Semifield metric spaces

1969 ◽  
Vol 1 (1) ◽  
pp. 127-136
Author(s):  
Martin Kleiber ◽  
W. J. Pervin
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Product Representation ◽  
Closed Set ◽  
Completely Regular ◽  
Closed Sets ◽  
Directed Set

Extending the results of An†onovskiĭ, Bol†janskiĭ, and Sarymsakov on semifield metric spaces, the authors define a regular semifield metric to be one in which the distance in the standard Tychonoff product representation of a point from a disjoint closed set is nonzero. It is shown that every completely regular topological space possesses a completely regular semifield metric and that there is an equivalent completely regular semifield metric for every semifield metric space. A normal semifield metric is defined to be one in which the distance between two disjoint closed sets is nonzero and it is shown that possessing a normal semifield metric is equivalent to being a normal topological space. Finally, Cauchy nets in semifield metric spaces are introduced leading to the notion of completeness. It is shown that a semifield metric space is complete iff every Cauchy net with the property that its directed set has cardinality less than or equal to the cardinality of the indexing set of the Tychonoff product representation of the semifield converges.

Nowhere Dense Subsets of Metric Spaces with Applications to Stone-Cech Compactifications

1972 ◽  
Vol 24 (4) ◽  
pp. 622-630 ◽  
Author(s):  
Jack R. Porter ◽  
R. Grant Woods
Keyword(s):  
Metric Space ◽  
Hausdorff Space ◽  
Metric Spaces ◽  
Dense Subset ◽  
Locally Compact ◽  
Completely Regular ◽  
Isolated Points ◽  
Remote Points ◽  
Topological Boundary ◽  
Regular Closed

Let X be a metric space. Assume either that X is locally compact or that X has no more than countably many isolated points. It is proved that if F is a nowhere dense subset of X, then it is regularly nowhere dense (in the sense of Katětov) and hence is contained in the topological boundary of some regular-closed subset of X. This result is used to obtain new properties of the remote points of the Stone-Čech compactification of a metric space without isolated points.Let βX denote the Stone-Čech compactification of the completely regular Hausdorff space X. Fine and Gillman [3] define a point p of βX to be remote if p is not in the βX-closure of a discrete subset of X.

scholarly journals Some Results on Wijsman Ideal Convergence in Intuitionistic Fuzzy Metric Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Ayhan Esi ◽  
Vakeel A. Khan ◽  
Mobeen Ahmad ◽  
Masood Alam
Keyword(s):  
Metric Space ◽  
Cauchy Sequence ◽  
Metric Spaces ◽  
General Setting ◽  
Fuzzy Metric Space ◽  
Ideal Convergence ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Closed Sets ◽  
Intuitionistic Fuzzy

In the present work, we study and extend the notion of Wijsman J –convergence and Wijsman J ∗ –convergence for the sequence of closed sets in a more general setting, i.e., in the intuitionistic fuzzy metric spaces (briefly, IFMS). Furthermore, we also examine the concept of Wijsman J ∗ –Cauchy and J –Cauchy sequence in the intuitionistic fuzzy metric space and observe some properties.

scholarly journals In which metric spaces are parallel bodies of closed sets closed?

1983 ◽  
Vol 26 (2) ◽  
pp. 265-273
Author(s):  
Gerald Beer
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Closed Ball ◽  
Closed Sets ◽  
Parallel Body

Let X be a metric space with metric d and for each x in X let Bλ[x] denote the closed ball of radius λ about x. Following Valentine [15] if K⊂X and λ is positive, then we call the set Bλ[K]=∪x∈KBλ[x] the λ-parallel body of K. The following fact is obvious.

scholarly journals On convergence of closed sets in a metric space and distance functions

1985 ◽  
Vol 31 (3) ◽  
pp. 421-432 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Distance Function ◽  
Metric Space ◽  
Pointwise Convergence ◽  
Distance Functions ◽  
Convergence Theorems ◽  
Closed Set ◽  
Closed Sets ◽  
The Family ◽  
Kuratowski Convergence ◽  
General Convergence

Let CL(X) denote the nonempty closed subsets of a metric space X. We answer the following question: in which spaces X is the Kuratowski convergence of a sequence {Cn} in CL(X) to a nonempty closed set C equivalent to the pointwise convergence of the distance functions for the sets in the sequence to the distance function for C ? We also obtain some related results from two general convergence theorems for equicontinuous families of real valued functions regarding the convergence of graphs and epigraphs of functions in the family.

scholarly journals Metric spaces with nice closed balls and distance functions for closed sets

1987 ◽  
Vol 35 (1) ◽  
pp. 81-96 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Distance Function ◽  
Metric Space ◽  
Pointwise Convergence ◽  
Metric Spaces ◽  
Closed Ball ◽  
Distance Functions ◽  
Entire Space ◽  
Closed Sets ◽  
Bounded Sets

A metric space 〈X,d〉 is said to have nice closed balls if each closed ball in X is either compact or the entire space. This class of spaces includes the metric spaces in which closed and bounded sets are compact and those for which the distance function is the zero-one metric. We show that these are the spaces in which the relation F = Lim Fn for sequences of closed sets is equivalent to the pointwise convergence of 〈d (.,Fn)〉 to d (.,F). We also reconcile these modes of convergence with three other closely related ones.

Descriptive sets

1966 ◽  
Vol 291 (1426) ◽  
pp. 353-367 ◽  
Keyword(s):  
Metric Space ◽  
Regular Space ◽  
Countable Union ◽  
Borel Set ◽  
Borel Sets ◽  
Countable Intersection ◽  
Completely Regular ◽  
Closed Sets ◽  
Open Set

A theory of descriptive Baire sets is developed for an arbitrary completely regular space. It is shown that descriptive Baire sets are Baire sets and that they form a system closed under countable union, countable intersection and intersection with a Baire set. If a descriptive Borel set (Rogers 1965) is a Baire set then it is a descriptive Baire set. If every open set is a countable union of closed sets, the descriptive Baire sets coincide with the descriptive Borel sets. It follows, in particular, that in a metric space a set is descriptive Baire, if, and only if, it is absolutely Borel and Lindelöf.

scholarly journals Fine Fuzzy sp Closed Sets in Fine Fuzzy Topological Spaces

2020 ◽  
Vol 9 (3) ◽  
pp. 1306-1313
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Inverse Image ◽  
Continuous Functions ◽  
Topological Spaces ◽  
The Novel ◽  
Closed Set ◽  
Closed Sets ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces

The main view of this article is the extended version of the fine topological space to the novel kind of space say fine fuzzy topological space which is developed by the notion called collection of quasi coincident of fuzzy sets. In this connection, fine fuzzy closed sets are introduced and studied some features on it. Further, the relationship between fine fuzzy closed sets with certain types of fine fuzzy closed sets are investigated and their converses need not be true are elucidated with necessary examples. Fine fuzzy continuous function is defined as the inverse image of fine fuzzy closed set is fine fuzzy closed and its interrelations with other types of fine fuzzy continuous functions are obtained. The reverse implication need not be true is proven with examples. Finally, the applications of fine fuzzy continuous function are explained by using the composition.

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.

Separating Closed Sets by Continuous Mappings into Developable Spaces

1981 ◽  
Vol 33 (6) ◽  
pp. 1420-1431 ◽  
Author(s):  
Harald Brandenburg
Keyword(s):  
Topological Space ◽  
Double Sequence ◽  
Completely Regular ◽  
Closed Sets ◽  
Continuous Mappings ◽  
Neighbourhood Base ◽  
Image Position ◽  
Metrizable Spaces ◽  
Metrization Theorem ◽  
Developable Spaces

A topological space X is called developable if it has a development, i.e., a sequence of open covers of X such that for each x ∈ X the collection is a neighbourhood base of x, whereThis class of spaces has turned out to be one of the most natural and useful generalizations of metrizable spaces [23]. In [4] it was shown that some well known results in metrization theory have counterparts in the theory of developable spaces (i.e., Urysohn's metrization theorem, the Nagata-Smirnov theorem, and Nagata's “double sequence theorem”). Moreover, in [3] it was pointed out that subspaces of products of developable spaces (i.e., D-completely regular spaces) can be characterized in much the same way as subspaces of products of metrizable spaces (i.e., completely regular T1-spaces).

scholarly journals Cyclic b -Multiplicative A , B -Hardy–Rogers-Type Local Contraction and Related Results in b -Multiplicative and b -Metric Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Abdullah Eqal Al-Mazrooei ◽  
Abdullah Shoaib ◽  
Jamshaid Ahmad
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Contractive Condition ◽  
Closed Set ◽  
Local Contraction ◽  
Multiplicative Metric Space

The aim of this paper is to define cyclic b -multiplicative Hardy–Rogers-type local contraction in the context of generalized spaces named as b -multiplicative spaces to extend various results of the literature including the main results of Yamaod et al. In this way, we apply a new generalized contractive condition only on a closed set instead of a whole set and by using b -multiplicative space instead of multiplicative metric space. We apply our results to obtain new results in b -metric spaces. Examples are given to show the usability of our results, when others cannot.

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