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On Finite and Infinite Principal Door and Connected-Topologies
ISSN: 1736-4337

Journal of Generalized Lie Theory and Applications
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  • Research Article   
  • J Generalized Lie Theory Appl, Vol 12(2): 291
  • DOI: 10.4172/2375-4435.1000291

On Finite and Infinite Principal Door and Connected-Topologies

Azzam AA1*, Farrag AS2 and EL-Sanousy E2
1Department of Mathematics, Faculty of Science, Assuit University, New Valley Branch, Egypt
2Department of Mathematics, Faculty of Science, Sohag University, Egypt
*Corresponding Author: Azzam AA, Department of Mathematics, Faculty of Science, Assuit University, New Valley Branch, Egypt, Tel: 0530867235, Email: [email protected]

Received Date: May 21, 2018 / Accepted Date: May 30, 2018 / Published Date: Jun 11, 2018

Abstract

A space X which carries topology τ is a door space if each subset of X is either open or closed. In this paper a characterization of the principle door and a formula for the number of the door topologies on a set Xn of n points are given. Some properties of the principal connected topologies on non-empty set X are discussed and the minimal τ0 -topologies on X are also characterized. Finally a few results about the number of the chain topologies on Xn are proved.

Keywords: Principal topological spaces; Door; Connected; Minimal τ0 ; Chain topologies AMS mathematics subject classification: 54D10, 54D15, 54C08 and 54C10

AMS mathematics subject classification

54D10, 54D15, 54C08 and 54C10

Introduction

Frohlich [1] defined the principal ultratopology on a set X to be the topology on X which is strictly weaker than the discrete topology D on X and which is of the form Dyz= Ez∪U(y), where U(y) is the principal ultra filter generated by {y} and Ez is the excluding point topology on X with the excluding point z. In fact Dyz=Ez∪Py, where Py is the particular point topology on X with the particular point y. Then, Dyz is the principal ultratopology on X in which each open set containing z contains y. Steiner [2] defined the minimal open set at a point x ∈ X in a space X which carries topology τ to be the open set Ux ∈ τ such that x ∈ Ux and is contained in each open set containing x. Steiner also defined the principal topology τ on X to be the topology with the minimal basis consists only of minimal open sets at the points of X, proved that τ is the principal if and only if arbitrary intersections of open sets are open and characterized the door topologies on X. In Farrag and Sewisy [3,4] and Farag and Abbas [5] described algorithms for construction and enumeration all strictly weaker topologies than a given topology on a set Xn of n points, all topologies and all hyperconnected, all door, connected and regular topologies on Xn.

Door Principal Topologies

Let X be a space which carries topology τ, Q and S be two any properties of topologies on X. Then, τ, is:

(1) An E-topology on X if ∪{G ∈ :GX}≠X [4].

(2) A P-topology on X if ∩{G ∈ :Gφ}≠φ [4].

(3) An h-topology on X if G∩Hφ for any G,H ∈ τ\{φ}. This is the irreducible [6] as well as is the hyperconnected [2]. If τ is finite then, h and P are equivalent but in general this is not true. For let, τ = {N,φ , N \{1, 2,3,...,n}: n∈N} where N is the set of the positive integers then, (N,τ) is h but not P.

(4) An E*-topology on X if there is a point p ∈ X such that Epτ where, Ep is the excluding point topology in X.

(5) An P*-topology on X if there is a point p ∈ X such that Ppτ, where Pp is the particular point topology on X with the particular point p ∈ X.

(6) An S(k)-topology on X if there are k singleton members of τ.

(7) A Q∨S-topology if it is Q or S.

(8) A Q∧S -topology if it is both Q and S.

(9) A Q\S -topology if it is Q and not S.

Throughout this paper |U| denote the cardinality of the set U. A finite set of n points is denoted by Xn and Nn, Nn(Q) denote the number of all topologies and the number of the Q -topologies on Xn. Then:

(1) equation where Nk is the number of all topologies on Xk [4].

(2) equation

Proposition 2.1

Let Xn be a set of n points then,

equation where equation

Proof. Let AXn be such that |A|=k,1≤kn. If τ is an E-topology on Xn\A then τ (A) = {G ∪ A:G∈τ }∪{φ} is a nondiscrete EP -topology on Xn. If A=Xn then {φ} is not a topology on Xn while equation Clearly there are nck nonempty subsets of Xn with the cardinality k and so,

equation

Secondly; if τ is a p-topology on Xn\A then equation is an E∧P-topology on Xn. If A=Xn then {φ} is not a topology on Xn while equation Similarly,

equation

Example 2.2: By using Example 3 [4] then,

equation

Remark 2.3: An E*-nondiscrete topology τ on a nonempty set X may be principal or nonprincipal. For, if X is an infinite set then, τ = {G ⊂ X : p∉G or X\G is finite} is a nonprincipal E*-topology on X.

Remark 2.4: A principal E*\E -topological space is not connected.

Remark 2.5: The P*-topologies on a set X are only principal since there is a point p ∈ X such that {x} = {p, x}∈τ and either

equation

Remark 2.6: A principal topology τ on a set X is E* if and only if τc is P* where, = equation

Remark 2.7: If β is the minimal basis for a P*-topology on a nonempty set X. Then, |U| is 1 or 2 for each U ∈ β and there is at least a point p ∈ X such that ∩{U ∈β :|U |= 2} = {p}∈τ . So, a -to equation pological space (X,τ) is not connected because G = ∪{U ∈β :|U |= 2} and equation are two nonempty members of τ.

Theorem 2.8: A principal topological space (X,τ) is door if and only if it is E*P*.

Proof. Clearly if τ is an E*P*-topology on a set X then it is door. Conversely; let (X,τ) be a door principal nonultra topological space and Ux be the minimal open set at the point x for each x ∈ X. Then, y ∈ Ux\{x} and X\{y} ∈ implies that equation which implies that equation because (X,τ) is a door space. Let p ∈ X be such that |Up| ≥ 3 then, x∈ X\{p} and |Ux| ≥ 2 implies that there is a point equation . Hence equation because equation because |Up| ≥ 3. If X \ {p,t}∈τ then equation which implies that t ∈ X \ {p,t} which is impossible. Hence equation which contradicts the assumption that (X,τ) is door. Therefore, Ux={x} for each x∈ X \ {p} which implies that (X,τ) is E*. If x ∈ X such that |Ux|=2, then, there is a point equation such that {p} ∈ τ. If x,y ∈ X are such that Ux≠Uy and equation Then, there are two points equation and equation . If q≠r then equation because equation and X \ {x,r}∈τ implies that equation implies that r ∈ X \ {x,r} which is impossible. Hence, equation which contradicts the assumption that (X,τ) is door. This contradiction implies that q=r=p. So, |Ux| is either 1 or 2 for each x ∈ X and such that

equation implies that equation this is if and only if equation So, (X,τ) is P*.

Corollary 2.9: A principal door topological space (X,τ) is connected if and only if it is EP∨PP where p ∈ X.

Remark 2.10: In previous study [7] proved that a door topological space (X,τ) is T0.

Clearly both the principal E* and P*-topological spaces are T0. It is T4 if τ is nonprincipal E* in which there is a point p ∈ X such that equation

Theorem 2.11: Let Xn be a set of n points then, equation

Proof. Let, β be the minimal basis for a nondiscrete E*-topology on Xn. Then, τ is S(n−1) and so there is a point p ∈ Xn such that equation and the member U ∈ β which is the minimal open set at the point p is such that |U| ≥ 2. If |U |= k ≥ 2, then U can be the minimal open set at each of its points i.e. p can be any point of U. Accordingly, we may have k distinct minimal bases β’s for E*-topologies on Xn. Since the number of such subsets U’s of Xn is nkc then the number of the corresponding distinct minimal bases for E*-topologies on Xn is k nkc . Therefore, 2 ≤ k ≤ n implies that: equation

Secondly; let β be the minimal basis for a nondiscrete P*-topology on Xn. Then there is a point p ∈ Xn such that {p} ∈ β and |U| is 1 or 2 for each U ∈ β such that equation . If T = {U ∈β :|U |= 2} and | T |= k −1≥1 then equation Clearly any of the points equation can take the position of the point p and we may have k distinct minimal bases for P*-topology on Xn. Since the number of the subsets equation of Xn is nkc , then the number of the corresponding minimal bases P*-topology on Xn is k nkc .

Therefore, 2 ≤ k ≤ n implies that:

equation

Clearly, equation and as a direct consequence of Theorems (2.7) and (2.10) we have Theorem (2.12).

Theorem 2.12: The number of all door topologies on Xn is:

equation

Proof. equation

Connected Principal-Topology

Let (X,τ) be a principle topological space, β be the minimal basis for τ and let T⊂β be such that equation and such that equation implies that equation then V ∈ τ is a minimal open set at each of its points since x ∈ V implies that Ux ∈ T and if G ∈ τ such that x ∈ G, then equation The family equation of such minimal open sets in non-empty and is a pair wise disjoints family of members of τ. Clearly,

(1) if equation for each T⊂β then each member of β is minimal at each of its points and by (X,τ) is regular [5].

(2) if equation for each T⊂β then there exists λ ∈ Δ such that equation which is the unique minimal open set at each of its points and (X,τ) is P. If equation . Then, equation for, if τ is an E-topology on X then Aλ=X for each λ∈ Δ. Otherwise let, x ∈ X, Tβ be such equation and equation such that equation implies that equation

If equation then equation and there is a point λ ∈ Δ such that U*=Vλ which implies that x ∈ A.

Theorem 3.1: Let τ be a principal topology on a set X, equation be the family of all open sets each of which is minimal at each of its points and equation for each λ ∈ Δ. If:

(a)V≠X, then (X,τ) is connected implies that A≠Vλ for any λ ∈ Δ.

(b) (X,τ) is connected then for each λ ∈ Δ there exists μ ∈ Δ such that equation

(c) equation for each two distinct points λ, μ ∈ Δ then (X,τ) is connected but not conversely.

Proof. (a): Let λ ∈ Δ be any point such that V≠X and x ∈ X be any point. Then equation implies that equation which implies by the definition of Aλ that x ∈ Aλ and if Aλ=Vλ, then x ∈ Vλ in fact Ux=Vλ. Therefore equation implies that x ∈ V and the contrapositive of this result is equation implies that equation implies that equation implies that (X,τ) is not connected. Therefore (X,τ) is connected implies that Aλ Vλ.

(b) Let, λ ∈ Δ be any point such that equation for each point μ ∈ Δ. Then A and X\Aλ are open sets which implies that (X,) is not connected. The contrapositive of this result, (X,) is connected implies that for each λ ∈ Δ there is μ ∈ Δ such that equation .

(c) Let (X,τ) be disconnected. Then there is a subset G of X such that equation . Then there is a point x ∈ G which implies that Ux⊂G and equation implies that there is a point λ ∈ Δ such that x ∈ A which implies by the definition of A that there is a point t ∈ X such that V⊂Ut and x ∈ Ut which implies that Ux⊂Ut. If

t ∈ X\G then equation which implies that equation a contradiction. Hence t ∈ G which implies that equation If equation then y ∈ Aλ implies that there is a point p ∈ X such that equation and equation implies that equation implies that equation while p ∈ G implies that y∈UP ⊂ G which implies that equation . Therefore equation which implies that A⊂G. Similarly there is a point μ ∈ Δ such that equation . Hence (X,τ) is not connected implies that there are two points equation such that equation and the contrapositive of this result is if equation for each two distinct points for each two distinct points λ,μ ∈Δ then (X,τ) is connected.

Conversely let, X={1,2,3,4,5,6,7} and β={{1}, {1,2,3}, {3}, {3,4,5}, {5}, {5,6,7,}, {7}} be the minimal basis for a topology τ on X. Then, (X,τ) is connected while A({1}) = {1,2,3} and A({7}) = {5,6,7} are disjoint.

Remark 3.2: If we denote the connected topologies on a set X by CTD-topologies. Then, the number of the connected topologies on a set Xn of n points is:

equation

Theorem 3.3: A principal T0-topological space (X,τ) is minimal T0 if and only if the minimal basis β for τ is totally ordered by the inclusion operator.

Proof. Let, (X,τ) be a T0-topological space such that the minimal basis β is totally ordered by the inclusion operator and τ* be a strictly weaker topology on X that τ. Then, by Theorem (2.8) [3] there are two distinct points y, z∈ X such that equation such that equation Then, equation because β is totally ordered by the inclusion operator which implies that equation is the minimal open set at both the points y and z in τyz which implies that (X,yz) is not T0. Therefore, (X,τ) is minimal T0.

Conversely; let (X,τ) be the minimal T0-topological space, y, z∈ X be two distinct points such that equation and equation If equation and G ∈ τyz such that z ∈ G then,

(1) yUz implies that equation

(2) equation implies that equation

(3) equation implies that G ∈ τ and

(4) z ∈ G implies that y ∈ G because G ∈ Dyz which implies that equation Hence, equation is the minimal open set at z. If equation then equation implies that either Ux=Uy or Ux=Uz which contradicts the assumption that equation and equation Hence, equation for each equation and equation implies that equation Then, (X,τyz) is T0 which contradicts that (X,τ) is minimal T0. This contradiction because of the incorrect assumption that equation and equation Therefore, either y ∈ Uz or z ∈ Uy for any two distinct points y, z∈ X . This completes the proof.

Corollary 3.4: Let, (X,τ) be T0 then (X,τyz) is T0 if and only if equation and equation for any two distinct points y, z∈ X where equation

Proof. As a direct consequence of the proof of Theorem(3.2) equation and equation implies that (X, yz ) is T0 for any two distinct points y, z∈ X .

Conversely; if (X,τyz) is T0 then, equation which implies that zUyand y ∈ Uz implies that τ=yz which implies that equation

Corollary 3.5: Let, (Xn,τ) be a minimal T0-topological space. then, there is a point p ∈ X such that equation .So, the number of the minimal T0-topologies on Xn is Nn (min. T0)=n!.

In the chain topology on a set Xn is the topology whose members are completely ordered by the inclusion operator. Clearly the minimal T0-topologies on Xn are chain topologies and the chain topologies on Xn are connected [8]. Stephen [8] proved that the number of all chain topologies on a set Xn is : equation . Where CHtopology on Xn is a chain topology on Xn and equation

The members of a chain topology τ on Xn are such that: equation in which G1 is nonempty and either singleton or nonsingleton. Accordingly τ is either S(1) or S(0) and so equation If 1 |G |= k and equation then τ is said to be CH(k)-topology on Xn and the number of the chain topologies in such case is denoted by equation . So, equation equation

Theorem 3.6: Let Xn be a set of n points then:

and

(4) equation where

N0 (CH) = 1.

Proof: Let A⊂Xn be such that | A|= k,1≤ k ≤ n . If τ is a chain topology on X n \A then, equation is an CH(k) -topology on Xn. Clearly there are nkc distinct nonempty subset of Xn with cardinality k and therefore:

(1) equation

(2) equation

Let k ∈ N and A⊂Xn be such that 1≤ k ≤ n −1 and | A|= r ≥ k . If τ is a CH(k)-topology on A then equation is a CH(k) -topology on Xn. If r=n then A⊂Xn and if τ is a CH(k)-topology on A then τ(A)=τ,

(3) equation

Clearly if k > n ≥ 0 then, equation and so equation . Also, if k=n then A=Xn which implies that τ={Xn,φ} which implies that equation which implies that equation . Therefore, using (1)

(4) equation

Conclusion

It is show that we are interested in finding the characterization of the principle door and a formula for the number of the door topologies on a set Xn of n points are given. Some properties of the principal connected topologies on a nonempty set X are discussed and the minimal T0-topologies on X are also characterized. Also, a few results about the number of the chain topologies on Xn are given.

References

Citation: Azzam AA, Farrag AS, EL-Sanousy E (2018) On Finite and Infinite Principal Door and Connected-Topologies. J Generalized Lie Theory Appl 12: 291. DOI: 10.4172/2375-4435.1000291

Copyright: © 2018 Azzam AA, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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