A note on discrete C-embedded subspaces

2019 ◽  
Vol 69 (2) ◽  
pp. 469-473 ◽  
Author(s):  
Mehrdad Namdari ◽  
Mohammad Ali Siavoshi
Keyword(s):  
Discrete Space ◽  
Countable Subset ◽  
Topological Spaces ◽  
Discrete Spaces ◽  
Measurable Cardinals ◽  
Hewitt Realcompactification

Abstract It is shown that in some non-discrete topological spaces, discrete subspaces with certain cardinality are C-embedded. In particular, this generalizes the well-known fact that every countable subset of P-spaces are C-embedded. In the presence of the measurable cardinals, we observe that if X is a discrete space then every subspace of υ X (i.e., the Hewitt realcompactification of X) whose cardinal is nonmeasurable, is a C-embedded, discrete realcompact subspace of υ X. This generalizes the well-known fact that the discrete spaces with nonmeasurable cardinal are realcompact.

scholarly journals Two preservation results for countable products of sequential spaces

2007 ◽  
Vol 17 (1) ◽  
pp. 161-172 ◽  
Author(s):  
MATTHIAS SCHRÖDER ◽  
ALEX SIMPSON
Keyword(s):  
Topological Spaces ◽  
Product Spaces ◽  
Convergence Spaces ◽  
Countable Product ◽  
Discrete Spaces ◽  
Sequential Spaces

We prove two results for the sequential topology on countable products of sequential topological spaces. First we show that a countable product of topological quotients yields a quotient map between the product spaces. Then we show that the reflection from sequential spaces to its subcategory of monotone ω-convergence spaces preserves countable products. These results are motivated by applications to the modelling of computation on non-discrete spaces.

scholarly journals N-compact spaces as limits of inverse systems of discrete spaces

1972 ◽  
Vol 14 (4) ◽  
pp. 467-469 ◽  
Author(s):  
Kim-Peu Chew
Keyword(s):  
Inverse Limit ◽  
Closed Subspace ◽  
Discrete Space ◽  
Natural Numbers ◽  
Compact Spaces ◽  
Finite Topology ◽  
Limit Space ◽  
Inverse Systems ◽  
Discrete Spaces ◽  
Inverse Limit Space

AbstractLet N denote the discrete space of all natural numbers. A space X is N-compact if it is homeomorphic with some closed subspace of a product of copies of N. In this paper, N-compact spaces are characterized as homeomorphs of inverse limit space of inverse systems of copies of subsets of N. Also, it is shown that a space X is N-compact if and only if the space (X) of all non-empty compact subsets of X with the finite topology is N-compact.

scholarly journals Geometric Discrete Unified Theory Framework

2020 ◽  
Author(s):  
Yan Breek
Keyword(s):  
High Stress ◽  
Higgs Field ◽  
Topological Defects ◽  
Wave Functions ◽  
Discrete Space ◽  
Topological Spaces ◽  
Unified Theory ◽  
Wave Particle Duality ◽  
Physical Forces ◽  
Theory Framework

This article proposes a unified theory framework encompassing a discrete topological interpretation of physical forces, wave functions, and the nature of space and time. It provides novel explanations for the collapse of wave functions, quantum entanglement, and offers insights into the origins of quantum probabilities. This article also explains the nature of mass, Higgs field, and suggests a path for unifying quantum mechanics and gravity. Elementary particles are represented as defects in discrete topological spaces. Entangled particles are directly connected to each other through a puncture in discrete space, separated by a distance of one Planck length. Wave functions are explained as mechanical stress waves within elastic discrete space. The results of the double-slit experiment are interpreted as wave functions maximizing the probability of rupture in high-stress areas of discrete space with obvious analogies to solid state mechanics. Wave-particle duality is explained as discrete topological defects causing extended distributed stress within space lattice.

Discrete resonances

2015 ◽  
Vol 29 (35n36) ◽  
pp. 1530016
Author(s):  
Franco Vivaldi
Keyword(s):  
Phase Space ◽  
Hamiltonian Systems ◽  
Discrete Space ◽  
Arithmetic Dynamics ◽  
Countable Number ◽  
Elliptic Orbits ◽  
Recent Developments ◽  
Linear And Nonlinear ◽  
Salient Features ◽  
Discrete Spaces

The concept of resonance has been instrumental to the study of Hamiltonian systems with divided phase space. One can also define such systems over discrete spaces, which have a finite or countable number of points, but in this new setting the notion of resonance must be re-considered from scratch. I review some recent developments in the area of arithmetic dynamics which outline some salient features of linear and nonlinear stable (elliptic) orbits over a discrete space, and also underline the difficulties that emerge in their analysis.

Some Classes of θ-Compactness

1986 ◽  
Vol 29 (1) ◽  
pp. 54-59
Author(s):  
S. Broverman
Keyword(s):  
Closed Subset ◽  
Closed Subspace ◽  
Discrete Space ◽  
Order Topology ◽  
Topological Spaces ◽  
Topological Product ◽  
Product Spaces ◽  
Bounded Space

AbstractLet A and A denote the classes of ordinal spaces with the order topology and Σ-product spaces of the two point discrete space respectively. Characterizations are given in terms of ultrafiIters of clopen sets of those O-dimensional Hausdorff topological spaces that can be embedded homeomorphically as a closed subspace of a topological product of either spaces from the class Λ or the class Δ. Both classes consist of spaces that are ω0-bounded. An example is given of a O-dimensional Hausdorff ω0-bounded space that cannot be homeomorphically embedded as a closed subset of a product of spaces from either Λ or Δ, answering a question of R. G. Woods.

First Countable Lindelöf Extensions of Uncountable Discrete Spaces

1980 ◽  
Vol 23 (4) ◽  
pp. 397-399 ◽  
Author(s):  
Murray Bell ◽  
John Ginsburg
Keyword(s):  
Discrete Space ◽  
Discrete Spaces

The existence of a first countable Lindelöf extension L of an uncountable discrete space D for which L-D is countable is considered. Assuming CH, such extensions exist; however it is also consistent that no such spaces exist, as follows from MA + ¬ CH

scholarly journals Pseudo-idempotents in semigroups of functions

1974 ◽  
Vol 18 (2) ◽  
pp. 182-187
Author(s):  
Frank A. Cezus
Keyword(s):  
Main Part ◽  
Continuous Functions ◽  
Discrete Space ◽  
Topological Spaces ◽  
Binary Relations ◽  
Generalize Theorem ◽  
Paper Form

The aim of this paper is to generalize Theorem 2.10 (i) of [2]. As stated in [2] this theorem deals with the semigroup of all selfmaps on a discrete space and provides a characterization of H-classes which contain an idempotent. We will generalize this theorem to the case of other semigroups of functions on a discrete space, some semigroups of continuous functions on non-discrete topological spaces, and one semigroup of binary relations. The results in this paper form the main part of chapter 3 of [1]. Some results will be quoted from [1] without proof; the required proofs can easily be supplied by the reader.

scholarly journals Mutually Compactificable Topological Spaces

2007 ◽  
Vol 2007 ◽  
pp. 1-10
Author(s):  
Martin Maria Kovár
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Discrete Space ◽  
The Other ◽  
Regular Space ◽  
Topological Spaces ◽  
Other Hand

Two disjoint topological spacesX,Yare(T2-)mutually compactificable if there exists a compact(T2-)topology onK=X∪Ywhich coincides onX,Ywith their original topologies such that the pointsx∈X,y∈Yhave open disjoint neighborhoods inK. This paper, the first one from a series, contains some initial investigations of the notion. Some key properties are the following: a topological space is mutually compactificable with some space if and only if it isθ-regular. A regular space on which every real-valued continuous function is constant is mutually compactificable with noS2-space. On the other hand, there exists a regular non-T3.5space which is mutually compactificable with the infinite countable discrete space.

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