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>

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.

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 Antisymmetry of the Stochastical Order on all Ordered Topological Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 250-252 ◽  
Author(s):  
Tobias Fritz
Keyword(s):  
Topological Space ◽  
General Result ◽  
Elementary Proof ◽  
Short Note ◽  
Stochastic Order ◽  
Topological Spaces ◽  
Probability Measures ◽  
Ordered Topological Space ◽  
Special Cases

Abstract In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.

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 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 theorem about countable decomposability

1982 ◽  
Vol 91 (3) ◽  
pp. 457-458 ◽  
Author(s):  
Roy O. Davies ◽  
Claude Tricot
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Continuous Functions ◽  
Perfect Set ◽  
Baire Function ◽  
Separable Metric Space

A function f:X → ℝ is countably decomposable (into continuous functions) if the topological space X can be partitioned into countably many sets An with each restriction f│ An continuous. According to L. V. Keldysh(2), the question whether every Baire function is countably decomposable was first raised by N. N. Luzin, and answered by P. S. Novikov. The answer is negative even for Baire-1 functions, as is shown in (2) (see also (1). In this paper we develop a characterization of the countably decomposable functions on a separable metric space X (see Corollary 1). We deduce that when X is complete they include all functions possessing the property P defined by D. E. Peek in (3): each non-empty σ-perfect set H contains a point at which f│ H is continuous. The example given by Peek shows that not every countably decomposable Baire-1 function has property P.

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.

scholarly journals Embeddings in the Fell and Wijsman topologies

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2747-2750
Author(s):  
Lubica Holá
Keyword(s):  
Topological Space ◽  
Partial Answer ◽  
Fell Topology ◽  
Wijsman Topology ◽  
Discrete Subspace

It is shown that if a T2 topological space X contains a closed uncountable discrete subspace, then the spaces (?1 + 1)? and (?1 + 1)?1 embed into (CL(X),?F), the hyperspace of nonempty closed subsets of X equipped with the Fell topology. If (X, d) is a non-separable perfect topological space, then (?1 + 1)? and (?1 +1)?1 embed into (CL(X), ?w(d)), the hyperspace of nonempty closed subsets of X equipped with the Wijsman topology, giving a partial answer to the Question 3.4 in [2].

Sign in / Sign up

Export Citation Format

Share Document