scholarly journals The smallest proper congruence on S(X)

1988 ◽  
Vol 30 (3) ◽  
pp. 301-313 ◽  
Author(s):  
K. H. Hofmann ◽  
K. D. Magill
Keyword(s):  
Topological Space ◽  
Complete Lattice ◽  
The Other ◽  
Topological Spaces ◽  
Other Hand ◽  
Dual Atom ◽  
Lattice Of Congruences

S(X) is the semigroup of all continuous self maps of the topological space X and for any semigroup S, Cong(S) will denote the complete lattice of congruences on S. Cong(S) has a zero Z and a unit U. Specifically, Z = {(a, a):a ∈ S} and U = S × S. Evidently, Z and U are distinct if S has at least two elements. By a proper congruence on S we mean any congruence which differs from each of these. Since S(X) has more than one element when X is nondegenerate, we will assume without further mention that the spaces we discuss in this paper have more than one point. We observed in [4] that there are a number of topological spaces X such that S(X) has a largest proper congruence, that is, Cong(S(X)) has a unique dual atom which is greater than every other proper congruence on S(X). On the other hand, we also found out in [5] that it is also common for S(X) to fail to have a largest proper congruence. We will see that the situation is quite different at the other end of the spectrum in that it is rather rare for S(X) not to have a smallest proper congruence. In other words, for most spaces X, Cong(S(X)) has a unique atom which is smaller than every other proper congruence.

scholarly journals Tilings in topological spaces

1999 ◽  
Vol 22 (3) ◽  
pp. 611-616 ◽  
Author(s):  
F. G. Arenas
Keyword(s):  
Topological Space ◽  
Pairwise Disjoint ◽  
The Other ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Topological Vector Spaces ◽  
Other Hand ◽  
Extensive Bibliography

Atilingof a topological spaceXis a covering ofXby sets (calledtiles) which are the closures of their pairwise-disjoint interiors. Tilings ofℝ2have received considerable attention (see [2] for a wealth of interesting examples and results as well as an extensive bibliography). On the other hand, the study of tilings of general topological spaces is just beginning (see [1, 3, 4, 6]). We give some generalizations for topological spaces of some results known for certain classes of tilings of topological vector spaces.

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.

Postnikov “Invariants” in 2004

2005 ◽  
Vol 12 (1) ◽  
pp. 139-155
Author(s):  
Julio Rubio ◽  
Francis Sergeraert
Keyword(s):  
Decision Problem ◽  
The Other ◽  
Topological Spaces ◽  
Exact Nature ◽  
Other Hand

Abstract The very nature of the so-called Postnikov invariants is carefully studied. Two functors, precisely defined, explain the exact nature of the connection between the category of topological spaces and the category of Postnikov towers. On one hand, these functors are in particular effective and lead to concrete machine computations through the general machine program Kenzo. On the other hand, the Postnikov “invariants” will be actual invariants only when an arithmetical decision problem – currently open – will be solved; it is even possible this problem is undecidable.

scholarly journals Rings with Boolean Lattices of One-Sided Annihilators

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1909
Author(s):  
Małgorzata Jastrzębska
Keyword(s):  
Complete Lattice ◽  
Associative Ring ◽  
The Other ◽  
Nilpotent Ring ◽  
Other Hand ◽  
Complemented Lattices ◽  
Reduced Ring ◽  
Boolean Lattices ◽  
Left And Right

The present paper is part of the research on the description of rings with a given property of the lattice of left (right) annihilators. The anti-isomorphism of lattices of left and right annihilators in any ring gives some kind of symmetry: the lattice of left annihilators is Boolean (complemented, distributive) if and only if the lattice of right annihilators is such. This allows us to restrict our investigations mainly to the left side. For a unital associative ring R, we prove that the lattice of left annihilators in R is Boolean if and only if R is a reduced ring. We also prove that the lattice of left annihilators of R being two-sided ideals is complemented if and only if this lattice is Boolean. The last statement, in turn, is known to be equivalent to the semiprimeness of R. On the other hand, for any complete lattice L, we construct a nilpotent ring whose lattice of left annihilators coincides with its sublattice of left annihilators being two-sided ideals and is isomorphic to L. This construction shows that the assumption of R being unital cannot be dropped in any of the above two results. Some additional results on rings with distributive or complemented lattices of left annihilators are obtained.

scholarly journals Weakly b-Closed Sets and Weakly b-Open Sets based of Fuzzy Neutrosophic bi-Topological Spaces

2020 ◽  
pp. 34-43
Author(s):  
Fatimah M. .. ◽  
◽  
◽  
Sarah W. Raheem
Keyword(s):  
The Other ◽  
Topological Spaces ◽  
Neutrosophic Sets ◽  
Basic Properties ◽  
Closed Sets ◽  
New Class ◽  
Other Hand ◽  
Definition Of

In this paper, we present and study some of the basic properties of the new class of sets called weakly b-closed sets and weakly b- open sets in fuzzy neutrosophic bi-topological spaces. We referred to some results related to the new definitions, which we taked the case of equal in the definition of b-sets instead of subset. Then, we discussed the relations between the new defined sets by hand and others fuzzy neutrosophic sets which were studied before us on the other hand on fuzzy neutrosophic bi-topological spaces. Then, we have studied some of characteristics and some relations are compared with necessary examples.

Topologies of practice

2011 ◽  
Vol 1 (3) ◽  
pp. 308-311 ◽  
Author(s):  
Mat Coleman
Keyword(s):  
Topological Space ◽  
Human Geography ◽  
The Other ◽  
Site Specific ◽  
Other Hand ◽  
The Difference ◽  
The One ◽  
A Site

Allen’s (2011) provocative argument on the difference between topographic and topologic ontologies in human geography offers human geographers an important opportunity to re-engage with other similarly spirited arguments about the limitations of the topographic. For example, debate over Marston et al.’s (2005) argument for a ‘site ontology’ has tended to sidestep the question of topological space and has instead dwelled on whether or not their representation of human geography research on scale is accurate. However, if Allen’s research gives human geographers another opportunity to take up the question of sociospatial practice as contingent, site-specific, and self-structuring, it also poses at least two problems. On the one hand, Allen characterizes the topographic and topologic according to a too neat calendar of sociospatial relations. On the other hand, Allen overlooks a long-standing appreciation for the topologic in human geography by drawing a strong distinction between past and newer intellectual approaches to power and space.

scholarly journals Pre Separation Axioms In Neutrosophic Crisp Topological Spaces

2020 ◽  
pp. 72-79
Author(s):  
Riad K. Al Al-Hamido ◽  
◽  
◽  
◽  
Luai Salha ◽  
...  
Keyword(s):  
Topological Space ◽  
The Other ◽  
Topological Spaces ◽  
Open Set ◽  
Separation Axioms ◽  
New Type

In this paper, A new type of separation axioms in the neutrosophic crisp Topological space named neutrosophic crisp pre separation axioms is going to be defined , in which neutrosophic crisp pre open set and neutrosophic crisp point are to be depended on. Also, relations among them and the other type are going to be found.

scholarly journals On (fuzzy) pseudo-semi-normed linear spaces

2021 ◽  
Vol 7 (1) ◽  
pp. 467-477
Author(s):  
Yaoqiang Wu ◽  
Keyword(s):  
The Other ◽  
Topological Spaces ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Other Hand ◽  
Fuzzy Topological Spaces

<abstract><p>In this paper, we introduce the notion of pseudo-semi-normed linear spaces, following the concept of pseudo-norm which was presented by Schaefer and Wolff, and illustrate their relationship. On the other hand, we introduce the concept of fuzzy pseudo-semi-norm, which is weaker than the notion of fuzzy pseudo-norm initiated by N$ \tilde{\rm{a}} $d$ \tilde{\rm{a}} $ban. Moreover, we give some examples which are according to the commonly used $ t $-norms. Finally, we establish norm structures of fuzzy pseudo-semi-normed spaces and provide (fuzzy) topological spaces induced by (fuzzy) pseudo-semi-norms, and prove that the (fuzzy) topological spaces are (fuzzy) Hausdorff.</p></abstract>

p-semisimple modules and type submodules

2019 ◽  
Vol 19 (04) ◽  
pp. 2050078
Author(s):  
A. Mozaffarikhah ◽  
E. Momtahan ◽  
A. R. Olfati ◽  
S. Safaeeyan
Keyword(s):  
Abelian Groups ◽  
Large Family ◽  
The Other ◽  
Topological Spaces ◽  
Algebraic Structures ◽  
Semisimple Modules ◽  
Other Hand ◽  
One To One ◽  
Regular Closed

In this paper, we introduce the concept of [Formula: see text]-semisimple modules. We prove that a multiplication reduced module is [Formula: see text]-semisimple if and only if it is a Baer module. We show that a large family of abelian groups are [Formula: see text]-semisimple. Furthermore, we give a topological characterizations of type submodules (ideals) of multiplication reduced modules ([Formula: see text]-semisimple rings). Moreover, we observe that there is a one-to-one correspondence between type ideals of some algebraic structures on one hand and regular closed subsets of some related topological spaces on the other hand. This also characterizes the form of closed ideals in [Formula: see text].

scholarly journals RegularL-fuzzy topological spaces and their topological modifications

2000 ◽  
Vol 23 (10) ◽  
pp. 687-695 ◽  
Author(s):  
T. Kubiak ◽  
M. A. de Prada Vicente
Keyword(s):  
Topological Space ◽  
The Other ◽  
Regular Space ◽  
Continuous Lattice ◽  
Topological Spaces ◽  
Scott Topology ◽  
Fuzzy Topological Spaces

ForLa continuous lattice with its Scott topology, the functorιLmakes every regularL-topological space into a regular space and so does the functorωLthe other way around. This has previously been known to hold in the restrictive class of the so-called weakly induced spaces. The concepts ofH-Lindelöfness (á la Hutton compactness) is introduced and characterized in terms of certain filters. RegularH-Lindelöf spaces are shown to be normal.

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