Separable Topological Space of Hereditary

NUTA Journal ◽  
2020 ◽  
Vol 7 (1-2) ◽  
pp. 68-70
Author(s):  
Raj Narayan Yadav ◽  
Bed Prasad Regmi ◽  
Surendra Raj Pathak
Keyword(s):  
Topological Space ◽  
Topological Property ◽  
Sufficient Condition ◽  
Topological Properties ◽  
Necessary And Sufficient Condition ◽  
Separable Space ◽  
Separable Topological Space ◽  
Necessary And Sufficient

A property of a topological space is termed hereditary ifand only if every subspace of a space with the property also has the property. The purpose of this article is to prove that the topological property of separable space is hereditary. In this paper we determine some topological properties which are hereditary and investigate necessary and sufficient condition functions for sub-spaces to possess properties of sub-spaces which are not in general hereditary.

Construction of sheaves by tolerance relations

2016 ◽  
Vol 09 (02) ◽  
pp. 1650035 ◽  
Author(s):  
M. P. K. Kishore ◽  
R. V. G. Ravi Kumar ◽  
P. Vamsi Sagar
Keyword(s):  
Topological Space ◽  
Universal Algebra ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

In this paper a method to construct a global sheaf space over a topological space for an arbitrary set using tolerance relations is proposed. It is observed that in general, the sheaf constructed by this method is different from the sheaf constructed by the method discussed in [U. M. Swamy, Representation of Universal algebra by sheaves, Proc. Amer. Math. Soc. 45 (1974) 55–58]. A necessary and sufficient condition for embedding a non-empty set into the set of all global sections of a sheaf over given topological space is established, and further an application over graphs is studied.

scholarly journals Intersections of Primary Ideals in Rings of Continuous Functions

1972 ◽  
Vol 24 (3) ◽  
pp. 502-519 ◽  
Author(s):  
R. Douglas Williams
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Prime Ideals ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Completely Regular ◽  
Rings Of Continuous Functions ◽  
Necessary And Sufficient ◽  
Primary Ideals

Let C be the ring of all real valued continuous functions on a completely regular topological space. This paper is an investigation of the ideals of C that are intersections of prime or of primary ideals.C. W. Kohls has analyzed the prime ideals of C in [3 ; 4] and the primary ideals of C in [5]. He showed that these ideals are absolutely convex. (An ideal I of C is called absolutely convex if |f| ≦ |g| and g ∈ I imply that f ∈ I.) It follows that any intersection of prime or of primary ideals is absolutely convex. We consider here the problem of finding a necessary and sufficient condition for an absolutely convex ideal I of C to be an intersection of prime ideals and the problem of finding a necessary and sufficient condition for I to be an intersection of primary ideals.

scholarly journals Minimal bases and minimal sub-bases for topological spaces

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 1957-1965
Author(s):  
Yiliang Li ◽  
Jinjin Li ◽  
Jun-e Feng ◽  
Hongkun Wang
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
General Topological Space ◽  
Minimal Base ◽  
Necessary And Sufficient

This paper investigates minimal bases and minimal sub-bases for topological spaces. First, a necessary and sufficient condition is derived for the existence of minimal base for a general topological space. Then the concept of minimal sub-base for a topological space is proposed and its properties are discussed. Finally, for Alexandroff spaces, some special results with respect to minimal bases and minimal sub-bases are illustrated.

scholarly journals Sequence spaces derived by the triple band generalized Fibonacci difference operator

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Taja Yaying ◽  
Bipan Hazarika ◽  
S. A. Mohiuddine ◽  
M. Mursaleen ◽  
Khursheed J. Ansari
Keyword(s):  
Hausdorff Measure ◽  
Difference Operator ◽  
Matrix Operator ◽  
Sufficient Condition ◽  
Topological Properties ◽  
Necessary And Sufficient Condition ◽  
Band Matrix ◽  
Necessary And Sufficient ◽  
Matrix Mappings ◽  
Triple Band

AbstractIn this article we introduce the generalized Fibonacci difference operator $\mathsf{F}(\mathsf{B})$ F ( B ) by the composition of a Fibonacci band matrix and a triple band matrix $\mathsf{B}(x,y,z)$ B ( x , y , z ) and study the spaces $\ell _{k}( \mathsf{F}(\mathsf{B}))$ ℓ k ( F ( B ) ) and $\ell _{\infty }(\mathsf{F}(\mathsf{B}))$ ℓ ∞ ( F ( B ) ) . We exhibit certain topological properties, construct a Schauder basis and determine the Köthe–Toeplitz duals of the new spaces. Furthermore, we characterize certain classes of matrix mappings from the spaces $\ell _{k}(\mathsf{F}(\mathsf{B}))$ ℓ k ( F ( B ) ) and $\ell _{\infty }(\mathsf{F}(\mathsf{B}))$ ℓ ∞ ( F ( B ) ) to space $\mathsf{Y}\in \{\ell _{\infty },c_{0},c,\ell _{1},cs_{0},cs,bs\}$ Y ∈ { ℓ ∞ , c 0 , c , ℓ 1 , c s 0 , c s , b s } and obtain the necessary and sufficient condition for a matrix operator to be compact from the spaces $\ell _{k}(\mathsf{F}(\mathsf{B}))$ ℓ k ( F ( B ) ) and $\ell _{\infty }(\mathsf{F}(\mathsf{B}))$ ℓ ∞ ( F ( B ) ) to $\mathsf{Y}\in \{ \ell _{\infty }, c, c_{0}, \ell _{1},cs_{0},cs,bs\} $ Y ∈ { ℓ ∞ , c , c 0 , ℓ 1 , c s 0 , c s , b s } using the Hausdorff measure of non-compactness.

scholarly journals Topological mixing of Weyl chamber flows

2020 ◽  
pp. 1-27
Author(s):  
NGUYEN-THI DANG ◽  
OLIVIER GLORIEUX
Keyword(s):  
Weyl Chamber ◽  
Sufficient Condition ◽  
Topological Properties ◽  
Necessary And Sufficient Condition ◽  
Right Action ◽  
Topological Mixing ◽  
The Right ◽  
Necessary And Sufficient

In this paper we study topological properties of the right action by translation of the Weyl chamber flow on the space of Weyl chambers. We obtain a necessary and sufficient condition for topological mixing.

scholarly journals Some Properties of Interior and Closure in General Topology

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 624
Author(s):  
Soon-Mo Jung ◽  
Doyun Nam
Keyword(s):  
Topological Space ◽  
Closed Subspace ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closed Set ◽  
Open Set ◽  
Necessary And Sufficient ◽  
Open Subspace

We present the necessary and sufficient conditions that the intersection of an open set and a closed set becomes either an open set or a closed set. As their dualities, we further introduce the necessary and sufficient conditions that the union of a closed set and an open set becomes either a closed set or an open set. Moreover, we give some necessary and sufficient conditions for the validity of U ∘ ∪ V ∘ = ( U ∪ V ) ∘ and U ¯ ∩ V ¯ = U ∩ V ¯ . Finally, we introduce a necessary and sufficient condition for an open subset of a closed subspace of a topological space to be open. As its duality, we also give a necessary and sufficient condition for a closed subset of an open subspace to be closed.

On Metrizability of Topological Spaces

1968 ◽  
Vol 20 ◽  
pp. 795-804 ◽  
Author(s):  
Carlos J. R. Borges
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Metrizable Space ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Moore Space ◽  
Necessary And Sufficient ◽  
Preceding Question

Our present work is divided into three sections. In §2 we study the metrizability of spaces with a Gδ-diagonal (see Definition 2.1). In §3 we study the metrization of topological spaces by means of collections of (not necessarily continuous) real-valued functions on a topological space. Our efforts, in §§2 and 3, are directed toward answering the following question: “Is every normal, metacompact (see Definition 2.4) Moore space a metrizable space?” which still remains unsolved. (However, Theorems 2.12 through 2.15 and Theorem 3.1 may be helpful in answering the preceding question.) In §4 we prove an apparently new necessary and sufficient condition for the metrizability of the Stone-Čech compactification of a metrizable space and hence for the compactness of a metric space.

Regularity of Richardson's Compactification

1974 ◽  
Vol 26 (6) ◽  
pp. 1289-1293 ◽  
Author(s):  
R. J. Gazik
Keyword(s):  
Topological Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Convergence Space ◽  
Regular Convergence ◽  
Hausdorff Convergence ◽  
Necessary And Sufficient

In [7] Richardson constructed a Stone-Čech type compactification R(E) of a Hausdorff convergence space E. Two questions arise in this regard. First, when is R(E) homeomorphic to β(E), β(E) the topological Stone-Čech compactification of E, for a Tychonoff topological space E? Second, if E is a regular convergence space, when is R(E) regular? The last question is motivated by the study of regular compactifications in [6]. In section 2 it will be shown that a necessary and sufficient condition in answer to both questions, is that α = cl(α) for each nonconvergent ultrafilter α on E.

scholarly journals FuzzyTL-uniform spaces

2006 ◽  
Vol 2006 ◽  
pp. 1-24 ◽  
Author(s):  
Khaled A. Hashem ◽  
Nehad N. Morsi
Keyword(s):  
Topological Space ◽  
Uniform Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Spaces ◽  
Fuzzy Topological Space ◽  
Necessary And Sufficient

The main purpose of this paper is to introduce a new structure that is a fuzzyTL-uniform space. We show that our structure generates a fuzzy topological space, precisely, a fuzzyT-locality space. Also, we deduce the concept of level uniformities of a fuzzyTL-uniformity. We connect the category of fuzzyTL-uniform spaces with the category of uniform spaces. We establish a necessary and sufficient condition, under which a fuzzyTL-uniformity is probabilistic pseudometrizable. Finally, we define a functor from the category of fuzzyTL-uniform spaces into the category of fuzzyT-locality spaces and we show that it preserves optimal lifts.

Ideal convergence of nets of functions with values in uniform spaces

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6281-6292
Author(s):  
Athanasios Megaritis
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Uniform Space ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Sets ◽  
Uniform Spaces ◽  
Ideal Convergence ◽  
Necessary And Sufficient

We consider the pointwise, uniform, quasi-uniform, and the almost uniform I-convergence for a net (fd)d?D of functions from a topological space X into a uniform space (Y,U), where I is an ideal on D. The purpose of the present paper is to provide ideal versions of some classical results and to extend these to nets of functions with values in uniform spaces. In particular, we define the notion of I-equicontinuous family of functions on which pointwise and uniform I-convergence coincide on compact sets. Generalizing the theorem of Arzel?, we give a necessary and sufficient condition for a net of continuous functions from a compact space into a uniform space to I-converge pointwise to a continuous function.

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