# On Finite and Infinite Principal Door and Connected-Topologies

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**Corresponding Author:**Azzam AA, Department of Mathematics, Faculty of Science, Assuit University, New Valley Branch, Egypt, Tel: 0530867235, Email: [email protected]

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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 D_{yz}= E_{z}∪U(y), where U(y) is the principal ultra filter generated by {y} and E_{z} is the excluding point topology on X with the excluding point z. In fact D_{yz}=E_{z}∪Py, where P_{y} is the particular point topology on X with the particular point y. Then, D_{yz} 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 U_{x} ∈ τ such that x ∈ U_{x} 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 X_{n} of n points, all topologies and all hyperconnected, all door, connected and regular topologies on X_{n}.

#### 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 E_{p}τ where, E_{p} is the excluding point topology in X.

(5) An P^{*}-topology on X if there is a point p ∈ X such that P_{p}τ, where P_{p }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 X_{n }and N_{n}, N_{n}(Q) denote the number of all topologies and the number of the Q -topologies on X_{n}. Then:

(1) where N_{k} is the number of all topologies on X_{k} [4].

(2)

**Proposition 2.1**

Let X_{n} be a set of n points then,

where

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

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

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

**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

**Remark 2.6:** A principal topology τ on a set X is E^{*} if and only if τ_{c} is P^{*} where, =

**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 pological space (X,τ) is not connected because G = ∪{U ∈β :|U |= 2} and 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 U

_{x}be the minimal open set at the point x for each x ∈ X. Then, y ∈ U

_{x}\{x} and X\{y} ∈ implies that which implies that 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 . Hence because because |Up| ≥ 3. If X \ {p,t}∈τ then which implies that t ∈ X \ {p,t} which is impossible. Hence which contradicts the assumption that (X,τ) is door. Therefore, U

_{x}={x} for each x∈ X \ {p} which implies that (X,τ) is E

^{*}. If x ∈ X such that |U

_{x}|=2, then, there is a point such that {p} ∈ τ. If x,y ∈ X are such that Ux≠Uy and Then, there are two points and . If q≠r then because and X \ {x,r}∈τ implies that implies that r ∈ X \ {x,r} which is impossible. Hence, which contradicts the assumption that (X,τ) is door. This contradiction implies that q=r=p. So, |U

_{x}| is either 1 or 2 for each x ∈ X and such that

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

**Corollary 2.9: **A principal door topological space (X,τ) is connected if and only if it is E_{P}∨P_{P} where p ∈ X.

**Remark 2.10: **In previous study [7] proved that a door topological space (X,τ) is T_{0}.

Clearly both the principal E^{*} and P^{*}-topological spaces are T_{0}. It is T_{4} if τ is nonprincipal E^{*} in which there is a point p ∈ X such that

**Theorem 2.11:** Let X_{n} be a set of n points then,

Proof. Let, β be the minimal basis for a nondiscrete E*-topology on X_{n}. Then, τ is S(n−1) and so there is a point p ∈ X_{n} such that 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 X_{n}. Since the number of such subsets U’s of X_{n} is ^{n}k_{c} then the number of the corresponding distinct minimal bases for E*-topologies on X_{n} is k ^{n}k_{c} . Therefore, 2 ≤ k ≤ n implies that:

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

Therefore, 2 ≤ k ≤ n implies that:

Clearly, 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 X_{n} is:

**Proof.**

#### 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 and such that implies that 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 The family of such minimal open sets in non-empty and is a pair wise disjoints family of members of τ. Clearly,

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

If then and there is a point λ ∈ Δ such that U^{*}=V_{λ} which implies that x ∈ A.

**Theorem 3.1:** Let τ be a principal topology on a set X, be the family of all open sets each of which is minimal at each of its points and 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

(c) 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 implies that which implies by the definition of A

_{λ}that x ∈ A

_{λ}and if A

_{λ}=V

_{λ}, then x ∈ V

_{λ }in fact U

_{x}=V

_{λ}. Therefore implies that x ∈ V and the contrapositive of this result is implies that implies that implies that (X,τ) is not connected. Therefore (X,τ) is connected implies that A

_{λ}V

_{λ}.

(b) Let, λ ∈ Δ be any point such that for each point μ ∈ Δ. Then A and X\A_{λ} 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 .

(c) Let (X,τ) be disconnected. Then there is a subset G of X such that . Then there is a point x ∈ G which implies that Ux⊂G and 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⊂U_{t} and x ∈ U_{t} which implies that U_{x}⊂U_{t}. If

t ∈ X\G then which implies that a contradiction. Hence t ∈ G which implies that If then y ∈ A_{λ} implies that there is a point p ∈ X such that and implies that implies that while p ∈ G implies that y∈U_{P} ⊂ G which implies that . Therefore which implies that A⊂G. Similarly there is a point μ ∈ Δ such that . Hence (X,τ) is not connected implies that there are two points such that and the contrapositive of this result is if 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 X_{n} of n points is:

**Theorem 3.3:** A principal T_{0}-topological space (X,τ) is minimal T_{0} if and only if the minimal basis β for τ is totally ordered by the inclusion operator.

* Proof.* Let, (X,τ) be a T

_{0}-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 such that Then, because β is totally ordered by the inclusion operator which implies that is the minimal open set at both the points y and z in τ

_{yz}which implies that (X,

_{yz}) is not T

_{0}. Therefore, (X,τ) is minimal T

_{0}.

Conversely; let (X,τ) be the minimal T_{0}-topological space, y, z∈ X be two distinct points such that and If and G ∈ τ_{yz} such that z ∈ G then,

(1) *yU _{z}* implies that

(2) implies that

(3) implies that G ∈ τ and

(4) z ∈ G implies that y ∈ G because G ∈ D_{yz} which implies that Hence, is the minimal open set at z. If then implies that either U_{x}=U_{y} or U_{x}=Uz which contradicts the assumption that and Hence, for each and implies that Then, (X,τ_{yz}) is T_{0} which contradicts that (X,τ) is minimal T_{0}. This contradiction because of the incorrect assumption that and Therefore, either y ∈ U_{z} or z ∈ U_{y} for any two distinct points y, z∈ X . This completes the proof.

**Corollary 3.4:** Let, (X,τ) be T_{0} then (X,τ** _{yz}**) is T

_{0}if and only if and for any two distinct points y, z∈ X where

Proof. As a direct consequence of the proof of Theorem(3.2) and implies that (X, _{yz} ) is T_{0} for any two distinct points y, z∈ X .

Conversely; if (X,τ_{yz}) is T_{0} then, which implies that *z*U* _{y}*and y ∈

*U*implies that τ=

_{z}_{yz}which implies that

**Corollary 3.5:** Let, (X_{n},τ) be a minimal T_{0}-topological space. then, there is a point p ∈ X such that .So, the number of the minimal T_{0}-topologies on X_{n} is Nn (min. T_{0})=n!.

In the chain topology on a set X_{n} is the topology whose members are completely ordered by the inclusion operator. Clearly the minimal T_{0}-topologies on X_{n} are chain topologies and the chain topologies on X_{n} are connected [8]. Stephen [8] proved that the number of all chain topologies on a set X_{n} is : . Where CHtopology on X_{n} is a chain topology on X_{n }and

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

**Theorem 3.6:** Let X_{n }be a set of n points then:

and

(4) where

*N _{0} (CH)* = 1.

* Proof: *Let A⊂X

_{n}be such that | A|= k,1≤ k ≤ n . If τ is a chain topology on

*X*then, is an C

_{n}\A*H(k)*-topology on X

_{n}. Clearly there are

^{n}k

_{c}distinct nonempty subset of X

_{n}with cardinality k and therefore:

(1)

(2)

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

(3)

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

(4)

#### 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 X_{n} of n points are given. Some properties of the principal connected topologies on a nonempty set X are discussed and the minimal T_{0}-topologies on X are also characterized. Also, a few results about the number of the chain topologies on X_{n }are given.

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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.