TWO SPACES OF MINIMAL PRIMES

2012 ◽  
Vol 11 (01) ◽  
pp. 1250014 ◽  
Author(s):  
PAPIYA BHATTACHARJEE
Keyword(s):  
Compact Space ◽  
Hausdorff Space ◽  
Topological Spaces ◽  
Zariski Topology ◽  
Topological Properties ◽  
Locally Compact ◽  
Inverse Topology ◽  
Minimal Prime ◽  
Stone Spaces ◽  
Prime Elements

This paper studies algebraic frames L and the set Min (L) of minimal prime elements of L. We will endow the set Min (L) with two well-known topologies, known as the Hull-kernel (or Zariski) topology and the inverse topology, and discuss several properties of these two spaces. It will be shown that Min (L) endowed with the Hull-kernel topology is a zero-dimensional, Hausdorff space; whereas, Min (L) endowed with the inverse topology is a T1, compact space. The main goal will be to find conditions on L for the spaces Min (L) and Min (L)-1 to have various topological properties; for example, compact, locally compact, Hausdorff, zero-dimensional, and extremally disconnected. We will also discuss when the two topological spaces are Boolean and Stone spaces.

A localic approach to minimal prime spectra

1988 ◽  
Vol 103 (1) ◽  
pp. 47-53 ◽  
Author(s):  
Sun Shu-Hao
Keyword(s):  
Distributive Lattice ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Topological Properties ◽  
Prime Spectrum ◽  
Minimal Prime

Throughout this paper, A will denote a distributive lattice with 0 and 1; we shall write spec A for the prime spectrum of A (i.e. the set of prime ideals of A, with the Stone–Zariski topology), and max A, min A for the subspaces of spec A consisting of maximal and minimal prime ideals respectively. These two subspaces have rather different topological properties: max A is always compact, but not always Hausdorff (indeed, any compact T1-space can occur as max A for some A), and min A is always Hausdorff (in fact zero-dimensional), but not always compact. (For more information on max A and min A, see Simmons[3].)

scholarly journals Isomorphisms from Extremely Regular Subspaces of C0K into C0S,X Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Manuel Felipe Cerpa-Torres ◽  
Michael A. Rincón-Villamizar
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Geometrical Parameter ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Continuous Image ◽  
Topological Properties ◽  
Locally Compact ◽  
Regular Subspace ◽  
Locally Compact Hausdorff Space

For a locally compact Hausdorff space K and a Banach space X, let C0K,X be the Banach space of all X-valued continuous functions defined on K, which vanish at infinite provided with the sup norm. If X is ℝ, we denote C0K,X as C0K. If AK be an extremely regular subspace of C0K and T:AK⟶C0S,X is an into isomorphism, what can be said about the set-theoretical or topological properties of K and S? Answering the question, we will prove that if X contains no copy of c0, then the cardinality of K is less than that of S. Moreover, if TT−1<3 and AK is also a subalgebra of C0K, the cardinality of the αth derivative of K is less than that of the αth derivative of S, for each ordinal α. Finally, if λX>1 and TT−1<λX, then K is a continuous image of a subspace of S. Here, λX is the geometrical parameter introduced by Jarosz in 1989: λX=infmaxx+λy:λ=1:x=y=1. As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.

scholarly journals ONE-POINT EXTENSIONS AND LOCAL TOPOLOGICAL PROPERTIES

2012 ◽  
Vol 88 (1) ◽  
pp. 12-16 ◽  
Author(s):  
M. R. KOUSHESH
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
General Question ◽  
Dense Subspace ◽  
Topological Properties ◽  
Locally Compact ◽  
Point Extension ◽  
One Point Compactification

AbstractA space $Y$ is called an extension of a space $X$ if $Y$ contains $X$ as a dense subspace. An extension $Y$ of $X$ is called a one-point extension of $X$ if $Y\setminus X$ is a singleton. P. Alexandroff proved that any locally compact non-compact Hausdorff space $X$ has a one-point compact Hausdorff extension, called the one-point compactification of $X$. Motivated by this, Mrówka and Tsai [‘On local topological properties. II’, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.19 (1971), 1035–1040] posed the following more general question: For what pairs of topological properties ${\mathscr P}$ and ${\mathscr Q}$ does a locally-${\mathscr P}$ space $X$ having ${\mathscr Q}$ possess a one-point extension having both ${\mathscr P}$ and ${\mathscr Q}$? Here, we provide an answer to this old question.

scholarly journals Locally compact space and continuity

BIBECHANA ◽  
1970 ◽  
Vol 7 ◽  
pp. 18-20
Author(s):  
Shitanshu Shekhar Choudhary ◽  
Raju Ram Thapa
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Compact Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Locally Compact ◽  
Locally Compact Space

Topological spaces for being T0, T1, T2 and regular space have been discussed. The conditions for a topological space to be locally compact have also been studied. We have found that a continuous function preserves locally compactness. Keywords: Topological spaces; Compactness; Regular space DOI: 10.3126/bibechana.v7i0.4038BIBECHANA 7 (2011) 18-20

scholarly journals On product of spaces of quasicomponents

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6307-6311
Author(s):  
Gjorgji Markoski ◽  
Abdulla Buklla
Keyword(s):  
Product Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Topological Properties ◽  
Locally Compact

We use a characterization of quasicomponents by continuous functions to obtain the well known theorem which states that product of quasicomponents Qx,Qy of topological spaces X,Y, respectively, gives quasicomponent in the product space X x Y. If spaces X,Y are locally-compact, paracompact and Haussdorf, then we prove that the space of quasicomponents of the product Q(XxY) is homeomorphic with the product space Q(X) x Q(Y), so these two spaces have the same topological properties.

scholarly journals Some Results in Neutrosophic Soft Topology Concerning Neutrosophic Soft ∗ b Open Sets

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Arif Mehmood ◽  
Saleem Abdullah ◽  
Mohammed M. Al-Shomrani ◽  
Muhammad Imran Khan ◽  
Orawit Thinnukool
Keyword(s):  
Compact Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Soft Topology ◽  
Open Set ◽  
Topological Features ◽  
Separation Axioms ◽  
Countable Space ◽  
Countably Compact

In this article, new generalised neutrosophic soft open known as neutrosophic soft ∗ b open set is introduced in neutrosophic soft topological spaces. Neutrosophic soft ∗ b open set is generated with the help of neutrosophic soft semiopen and neutrosophic soft preopen sets. Then, with the application of this new definition, some soft neutrosophical separation axioms, countability theorems, and countable space can be Hausdorff space under the subjection of neutrosophic soft sequence which is convergent, the cardinality of neutrosophic soft countable space, engagement of neutrosophic soft countable and uncountable spaces, neutrosophic soft topological features of the various spaces, soft neutrosophical continuity, the product of different soft neutrosophical spaces, and neutrosophic soft countably compact that has the characteristics of Bolzano Weierstrass Property (BVP) are studied. In addition to this, BVP shifting from one space to another through neutrosophic soft continuous functions, neutrosophic soft sequence convergence, and its marriage with neutrosophic soft compact space, sequentially compactness are addressed.

Zariski topology on lattice modules

2015 ◽  
Vol 08 (04) ◽  
pp. 1550066 ◽  
Author(s):  
Sachin Ballal ◽  
Vilas Kharat
Keyword(s):  
New York ◽  
Spectral Theory ◽  
Zariski Topology ◽  
Topological Properties ◽  
Algebraic Properties ◽  
Prime Elements ◽  
Multiplicative Lattices

Let [Formula: see text] be a lattice module over a [Formula: see text]-lattice [Formula: see text] and [Formula: see text] be the set of all prime elements in lattice modules [Formula: see text]. In this paper, we study the generalization of the Zariski topology of multiplicative lattices [N. K. Thakare, C. S. Manjarekar and S. Maeda, Abstract spectral theory II: Minimal characters and minimal spectrums of multiplicative lattices, Acta Sci. Math. 52 (1988) 53–67; N. K. Thakare and C. S. Manjarekar, Abstract spectral theory: Multiplicative lattices in which every character is contained in a unique maximal character, in Algebra and Its Applications (Marcel Dekker, New York, 1984), pp. 265–276.] to lattice modules. Also we investigate the interplay between the topological properties of [Formula: see text] and algebraic properties of [Formula: see text].

Concerning σ-Connectedness of Baire Spaces

1980 ◽  
Vol 32 (6) ◽  
pp. 1438-1447
Author(s):  
A. García-Máynez
Keyword(s):  
Compact Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Compact Sets ◽  
Connected Space ◽  
Locally Compact ◽  
Whole Space ◽  
Locally Compact Hausdorff Space

A well known theorem of Sierpiński states that every compact connected Hausdorff space is σ-connected. Hence, if X is locally compact and Hausdorff and X is locally connected at x, then x has a σ-connected neighborhood. However, local connectedness at x is not a necessary condition for x to have a σ-connected neighborhood, because the whole space may be σ-connected without being locally connected at x. One of the purposes of the present paper is then to investigate which points of a given locally compact Hausdorff space have σ-connected neighborhoods. We find also sufficient conditions for a connected, hereditarily Baire space to be σ-connected and prove the impossibility of expressing a connected, Čech-complete, rim compact space as a countable infinite union of mutually disjoint compact sets. Finally, we introduce the concept of D-connected space and relate it to σ-connectedness.

Semi-Groups of Maps in a Locally Compact Space

1967 ◽  
Vol 19 ◽  
pp. 688-696 ◽  
Author(s):  
J. R. Dorroh
Keyword(s):  
Compact Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
List Type ◽  
Type Number ◽  
Semi Group ◽  
Locally Compact ◽  
Locally Compact Space ◽  
Locally Compact Hausdorff Space

Suppose that S is a locally compact Hausdorff space. A one-parameter semi-group of maps in S is a family {ϕt; t ⩾ 0} of continuous functions from S into S satisfying(i)ϕt0ϕu = ϕt+u for t, u ⩾ 0, where the circle denotes composition, and(ii)ϕ0 = e, the identity map on S.A semi-group {ϕt} of maps in S is said to be(iii)of class (C0) if ϕt(x) → x as t → 0 for each x in S,(iv)separately continuous if the function t → ϕt(x) is continuous on [0, ∞) for each x in S, and(v)doubly continuous if the function (t, x) → (ϕt(x) is continuous on [0, ∞) x S.

scholarly journals THE LOCAL DENSITY AND THE LOCAL WEAK DENSITY IN THE SPACE OF PERMUTATION DEGREE AND IN HATTORI SPACE

2020 ◽  
Vol 6 (2) ◽  
pp. 108
Author(s):  
Tursun K. Yuldashev ◽  
Farhod G. Mukhamadiev
Keyword(s):  
Positive Integer ◽  
Permutation Group ◽  
Real Line ◽  
Local Density ◽  
Cardinal Number ◽  
Topological Spaces ◽  
Topological Properties ◽  
Locally Compact ◽  
The Real ◽  
Weak Density

In this paper, the local density \((l d)\) and the local weak density \((l w d)\) in the space of permutation degree as well as the cardinal and topological properties of Hattori spaces are studied. In other words, we study the properties of the functor of permutation degree \(S P^{n}\) and the subfunctor of permutation degree \(S P_{G}^{n}\),  \(P\) is the cardinal number of topological spaces. Let \(X\) be an infinite \(T_{1}\)-space. We prove that the following propositions hold.(1) Let \(Y^{n} \subset X^{n}\); (A) if \(d\, \left(Y^{n} \right)=d\, \left(X^{n} \right)\), then \(d\, \left(S P^{n} Y\right)=d\, \left(SP^{n} X\right)\); (B) if \(l w d\, \left(Y^{n} \right)=l w d\, \left(X^{n} \right)\), then \(l w d\, \left(S P^{n} Y\right)=l w d\, \left(S P^{n} X\right)\). (2) Let \(Y\subset X\); (A) if \(l d \,(Y)=l d \,(X)\), then \(l d\, \left(S P^{n} Y\right)=l d\, \left(S P^{n} X\right)\); (B) if \(w d \,(Y)=w d \,(X)\), then \(w d\, \left(S P^{n} Y\right)=w d\, \left(S P^{n} X\right)\).(3) Let \(n\) be a positive integer, and let \(G\) be a subgroup of the permutation group \(S_{n}\). If \(X\) is a locally compact \(T_{1}\)-space, then \(S P^{n} X, \, S P_{G}^{n} X\), and \(\exp _{n} X\) are \(k\)-spaces.(4) Let \(n\) be a positive integer, and let \(G\) be a subgroup of the permutation group \(S_{n}\). If \(X\) is an infinite \(T_{1}\)-space, then \(n \,\pi \,w \left(X\right)=n \, \pi \,w \left(S P^{n} X \right)=n \,\pi \,w \left(S P_{G}^{n} X \right)=n \,\pi \,w \left(\exp _{n} X \right)\).We also have studied that the functors \(SP^{n},\) \(SP_{G}^{n} ,\) and \(\exp _{n} \) preserve any \(k\)-space. The functors \(SP^{2}\) and \(SP_{G}^{3}\) do not preserve Hattori spaces on the real line. Besides, it is proved that the density of an infinite \(T_{1}\)-space \(X\) coincides with the densities of the spaces \(X^{n}\), \(\,S P^{n} X\), and \(\exp _{n} X\). It is also shown that the weak density of an infinite \(T_{1}\)-space \(X\) coincides with the weak densities of the spaces \(X^{n}\), \(\,S P^{n} X\), and \(\exp _{n} X\).

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