scholarly journals Remarks on γˆ*Ω-open sets and minimal γˆ*Ω-open sets γˆ*Ω

2019 ◽  
Vol 9 (1S4) ◽  
pp. 1017-1021
Keyword(s):  
Topological Space ◽  
Locally Finite ◽  
Basic Properties ◽  
Finite Space ◽  
The Impact

Aim of this paper is to define -open sets in a topological space and obtain their basic properties. Also, we define minimal -open sets in a space and study the impact of two minimal -open sets in a space with -regular operation. However, the roll of minimal -open sets in -locally finite space has been discussed.

More on Extending Continuous Pseudometrics

1970 ◽  
Vol 22 (5) ◽  
pp. 984-993 ◽  
Author(s):  
H. L. Shapiro
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Closed Subset ◽  
Locally Finite ◽  
Whole Space

The concept of extending to a topological space X a continuous pseudometric defined on a subspace S of X has been shown to be very useful. This problem was first studied by Hausdorff for the metric case in 1930 [9]. Hausdorff showed that a continuous metric on a closed subset of a metric space can be extended to a continuous metric on the whole space. Bing [4] and Arens [3] rediscovered this result independently. Recently, Shapiro [15] and Alo and Shapiro [1] studied various embeddings. It has been shown that extending pseudometrics can be characterized in terms of extending refinements of various types of open covers. In this paper we continue our study of extending pseudometrics. First we show that extending pseudometrics can be characterized in terms of σ-locally finite and σ-discrete covers. We then investigate when can certain types of covers be extended.

A Note on Extending Locally Finite Collections

1976 ◽  
Vol 19 (1) ◽  
pp. 117-119
Author(s):  
H. L. Shapiro ◽  
F. A. Smith
Keyword(s):  
Topological Space ◽  
Set Cover ◽  
Topological Spaces ◽  
Locally Finite ◽  
Whole Space

Recently there has been a great deal of interest in extending refinements of locally finite and point finite collections on subsets of certain topological spaces. In particular the first named author showed that a subset S of a topological space X is P-embedded in X if and only if every locally finite cozero-set cover on S has a refinement that can be extended to a locally finite cozero-set cover of X. Since then many authors have studied similar types of embeddings (see [1], [2], [3], [4], [6], [8], [9], [10], [11], and [12]). Since the above characterization of P-embedding is equivalent to extending continuous pseudometrics from the subspace S up to the whole space X, it is natural to wonder when can a locally finite or a point finite open or cozero-set cover on S be extended to a locally finite or point-finite open or cozero-set cover on X.

scholarly journals Paracompactness with respect to an ideal

1997 ◽  
Vol 20 (3) ◽  
pp. 433-442 ◽  
Author(s):  
T. R. Hamlett ◽  
David Rose ◽  
Dragan Janković
Keyword(s):  
Topological Space ◽  
Finite Union ◽  
Open Cover ◽  
Locally Finite ◽  
Open Set ◽  
Special Cases ◽  
Paracompact Spaces

An ideal on a setXis a nonempty collection of subsets ofXclosed under the operations of subset and finite union. Given a topological spaceXand an idealℐof subsets ofX,Xis defined to beℐ-paracompact if every open cover of the space admits a locally finite open refinement which is a cover for all ofXexcept for a set inℐ. Basic results are investigated, particularly with regard to theℐ- paracompactness of two associated topologies generated by sets of the formU−IwhereUis open andI∈ℐand⋃{U|Uis open andU−A∈ℐ, for some open setA}. Preservation ofℐ-paracompactness by functions, subsets, and products is investigated. Important special cases ofℐ-paracompact spaces are the usual paracompact spaces and the almost paracompact spaces of Singal and Arya [“On m-paracompact spaces”, Math. Ann., 181 (1969), 119-133].

scholarly journals The Spectral Sequence of a Covering

1961 ◽  
Vol 12 (3) ◽  
pp. 149-158 ◽  
Author(s):  
D. J. Simms
Keyword(s):  
Topological Space ◽  
Spectral Sequence ◽  
Abelian Groups ◽  
Locally Finite

Let be a covering of a topological space X and ℱ a sheaf of abelian groups over X. By a well known result of Leray, (3) II theorems 5.2.4. and 5.4.1., if is open, or closed and locally finite, there exists a spectral sequence {Er} satisfying isomorphisms and for some filtration of the graded group H*(X, ℱ). ℋq(ℱ) denotes the system of coefficients over : s→Hq(| s |, ℱ).

A View on Fuzzy Minimal Open Sets and Fuzzy Maximal Open Sets

2015 ◽  
Vol 7 (1) ◽  
pp. 62-73 ◽  
Author(s):  
Hamid Reza Moradi
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Basic Properties ◽  
New Class ◽  
Properties And Characterization ◽  
Open Set ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces ◽  
Characterization Theorems

A nonzero fuzzy open set () of a fuzzy topological space is said to be fuzzy minimal open (resp. fuzzy maximal open) set if any fuzzy open set which is contained (resp. contains) in is either or itself (resp. either or itself). In this note, a new class of sets called fuzzy minimal open sets and fuzzy maximal open sets in fuzzy topological spaces are introduced and studied which are subclasses of open sets. Some basic properties and characterization theorems are also to be investigated.

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.

scholarly journals Basic Properties of Metrizable Topological Spaces

2009 ◽  
Vol 17 (3) ◽  
pp. 201-205 ◽  
Author(s):  
Karol Pąk
Keyword(s):  
Topological Space ◽  
General Topology ◽  
Topological Spaces ◽  
Basic Properties ◽  
Special Case

Basic Properties of Metrizable Topological Spaces We continue Mizar formalization of general topology according to the book [11] by Engelking. In the article, we present the final theorem of Section 4.1. Namely, the paper includes the formalization of theorems on the correspondence between the cardinalities of the basis and of some open subcover, and a discreet (closed) subspaces, and the weight of that metrizable topological space. We also define Lindelöf spaces and state the above theorem in this special case. We also introduce the concept of separation among two subsets (see [12]).

scholarly journals Bombay hypertopologies

2003 ◽  
Vol 4 (2) ◽  
pp. 421 ◽  
Author(s):  
Giuseppe Di Maio ◽  
Enrico Meccariello ◽  
Somashekhar Naimpally
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Wijsman Convergence ◽  
Locally Finite ◽  
Simple Result ◽  
Special Cases ◽  
Wijsman Topology

<p>Recently it was shown that, in a metric space, the upper Wijsman convergence can be topologized with the introduction of a new far-miss topology. The resulting Wijsman topology is a mixture of the ball topology and the proximal ball topology. It leads easily to the generalized or g-Wijsman topology on the hyperspace of any topological space with a compatible LO-proximity and a cobase (i.e. a family of closed subsets which is closed under finite unions and which contains all singletons). Further generalization involving a topological space with two compatible LO-proximities and a cobase results in a new hypertopology which we call the Bombay topology. The generalized locally finite Bombay topology includes the known hypertopologies as special cases and moreover it gives birth to many new hypertopologies. We show how it facilitates comparison of any two hypertopologies by proving one simple result of which most of the existing results are easy consequences.</p>

scholarly journals On a notion of entropy in coarse geometry

2019 ◽  
Vol 7 (1) ◽  
pp. 48-68
Author(s):  
Nicolò Zava
Keyword(s):  
Large Scale ◽  
Topological Spaces ◽  
Coarse Geometry ◽  
Locally Finite ◽  
Basic Properties ◽  
Algebraic Entropy ◽  
Probability Spaces ◽  
Scale Properties ◽  
Coarse Space

AbstractThe notion of entropy appears in many branches of mathematics. In each setting (e.g., probability spaces, sets, topological spaces) entropy is a non-negative real-valued function measuring the randomness and disorder that a self-morphism creates. In this paper we propose a notion of entropy, called coarse entropy, in coarse geometry, which is the study of large-scale properties of spaces. Coarse entropy is defined on every bornologous self-map of a locally finite quasi-coarse space (a recent generalisation of the notion of coarse space, introduced by Roe). In this paper we describe this new concept, providing basic properties, examples and comparisons with other entropies, in particular with the algebraic entropy of endomorphisms of monoids.

Sign in / Sign up

Export Citation Format

Share Document