scholarly journals Semi- I -Expandable Ideal Topological Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Chawalit Boonpok
Keyword(s):  
Topological Spaces ◽  
Locally Finite

Our purpose is to introduce the notion of semi- I -expandable ideal topological spaces. Some properties of semi- I -locally finite collections are investigated. In particular, several characterizations of semi- I -expandable ideal topological spaces are established.

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.

Finite posets and topological spaces in locally finite varieties

2005 ◽  
Vol 52 (2-3) ◽  
pp. 119-136 ◽  
Author(s):  
Benoit Larose ◽  
L�szl� Z�dori
Keyword(s):  
Topological Spaces ◽  
Locally Finite ◽  
Locally Finite Varieties ◽  
Finite Posets

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.

scholarly journals Some applications of minimal open sets

2001 ◽  
Vol 27 (8) ◽  
pp. 471-476 ◽  
Author(s):  
Fumie Nakaoka ◽  
Nobuyuki Oda
Keyword(s):  
Hausdorff Space ◽  
Nonempty Subset ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Locally Finite ◽  
Open Set ◽  
Finite Space

We characterize minimal open sets in topological spaces. We show that any nonempty subset of a minimal open set is pre-open. As an application of a theory of minimal open sets, we obtain a sufficient condition for a locally finite space to be a pre-Hausdorff space.

scholarly journals Decomposition, Mapping, and Sum Theorems of ω-Paracompact Topological Spaces

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 339
Author(s):  
Samer Al Al Ghour
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Locally Finite ◽  
Open Questions ◽  
Locally Countable ◽  
Sum Theorem

As a weaker form of ω-paracompactness, the notion of σ-ω-paracompactness is introduced. Furthermore, as a weaker form of σ-ω-paracompactness, the notion of feebly ω-paracompactness is introduced. It is proven hereinthat locally countable topological spaces are feebly ω-paracompact. Furthermore, it is proven hereinthat countably ω-paracompact σ-ω-paracompact topological spaces are ω-paracompact. Furthermore, it is proven hereinthat σ-ω-paracompactness is inverse invariant under perfect mappings with countable fibers, and as a result, is proven hereinthat ω-paracompactness is inverse invariant under perfect mappings with countable fibers. Furthermore, if A is a locally finite closed covering of a topological space X,τ with each A∈A being ω-paracompact and normal, then X,τ is ω-paracompact and normal, and as a corollary, a sum theorem for ω-paracompact normal topological spaces follows. Moreover, three open questions are raised.

scholarly journals Some Applications of Minimal Pγ -Open Sets

2014 ◽  
Vol 47 (2) ◽  
Author(s):  
Alias B. Khalaf ◽  
Hariwan Z. Ibrahim
Keyword(s):  
Hausdorff Space ◽  
Nonempty Subset ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Locally Finite ◽  
Open Set ◽  
Finite Space

AbstractWe characterize minimal P γ-open sets in topological spaces. We show that any nonempty subset of a minimal P γ-open set is pre P-open. As an application of a theory of minimal Pγ-open sets, we obtain a sufficient condition for a P-locally finite space to be a pre P γ-Hausdorff space

RANDOM DYNAMICAL SYSTEMS ON ORDERED TOPOLOGICAL SPACES

2006 ◽  
Vol 06 (03) ◽  
pp. 255-300 ◽  
Author(s):  
HANS G. KELLERER ◽  
G. WINKLER
Keyword(s):  
Invariant Measure ◽  
Topological Spaces ◽  
Random Dynamical System ◽  
Locally Finite ◽  
Large Numbers ◽  
Markov Chain Theory ◽  
Null Recurrence ◽  
Finite Invariant Measure ◽  
Positive Recurrent ◽  
Discrete Markov Chain

Let (Xn, n ≥ 0) be a random dynamical system and its state space be endowed with a reasonable topology. Instead of completing the structure as common by some linearity, this study stresses — motivated in particular by economic applications — order aspects. If the underlying random transformations are supposed to be order-preserving, this results in a fairly complete theory. First of all, the classical notions of and familiar criteria for recurrence and transience can be extended from discrete Markov chain theory. The most important fact is provided by the existence and uniqueness of a locally finite-invariant measure for recurrent systems. It allows to derive ergodic theorems as well as to introduce an attract or in a natural way. The classification is completed by distinguishing positive and null recurrence corresponding, respectively, to the case of a finite or infinite invariant measure; equivalently, this amounts to finite or infinite mean passage times. For positive recurrent systems, moreover, strengthened versions of weak convergence as well as generalized laws of large numbers are available.

scholarly journals Properties of space set topological spaces

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2475-2487 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Topological Space ◽  
Crucial Role ◽  
Topological Structure ◽  
Special Kind ◽  
Topological Spaces ◽  
Separation Axiom ◽  
Locally Finite ◽  
Alexandroff Space ◽  
Finite Spaces ◽  
Point Set

Since a locally finite topological structure plays an important role in the fields of pure and applied topology, the paper studies a special kind of locally finite spaces, so called a space set topology (for brevity, SST) and further, proves that an SST is an Alexandroff space satisfying the separation axiom T0. Unlike a point set topology, since each element of an SST is a space, the present paper names the topology by the space set topology. Besides, for a connected topological space (X,T) with |X| = 2 the axioms T0, semi-T1/2 and T1/2 are proved to be equivalent to each other. Furthermore, the paper shows that an SST can be used for studying both continuous and digital spaces so that it plays a crucial role in both classical and digital topology, combinatorial, discrete and computational geometry. In addition, a connected SST can be a good example showing that the separation axiom semi-T1/2 does not imply T1/2.

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