Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

ON FINITE T_{0} TOPOLOGICAL SPACES

A. EL-FATTAH EL-ATIK, M. E. ABD EL-MONSEF, AND E. I. LASHIN

Abstract. Finite topological spaces became much more essential in topology, with the development of computer science. The task of this paper is to study and investigate some properties of such spaces with the existence of an ordered relation between their minimal neighborhoods. We introduce notations and elementary facts known as Alexandroff space [4, 3, 5]. The family of minimal neighborhoods forms a unique minimal base. We consider T_{0} spaces. We give a link between finite T_{0} spaces and the related partial order. Finally, we study some properties of multifunctions and their relationships with connected ordered topological spaces.

1. Introduction and Preliminaries

Finite spaces were first studied by P.A. Alexandroff in 1937 in [1]. Actually, finite spaces had been more earlier investigated by many authors under the name of simplicial complexes. There were several other contributions by Flachsmeyer in 1961 [12], Stong in 1966 [22] and L. Lotz in 1970 [19]. Rinow [21] in his book discussed some properties of finite spaces. However, the subject has never been considered as a main field of topology.

With the progress of computer technology, finite spaces have become more important. Herman in 1990 [13], Khalimsky and et. al. in 1990 [14], Kong and Kopperman in 1991 [15] have been applied them to model the computer screen. In this paper we focus on finite spaces with order. The main importance of our study is to offer a new formulations for some topological operators in general topology such as interior, closure, boundary and exterior operators. We present and study comparisons between some topological properties in the case of finite spaces. In what follows, by X we mean always

2000 Mathematics Subject Classification. Primary: 54B10, 54D30; Secondary: 54A05, 54G99.

Key words and phrases. T_{0} spaces, finite T_{0} spaces, minimal neighborhoods, partially ordered sets, multifunctions, topological dimension, weak continuity.

A. El-Fattah El-Atik, M. E. Abd El-Monsef and E. I. Lashin, On finite T_{0} topological spaces, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 75– 90, Topology Atlas, Toronto, 2002; math.arXiv:GN/0204123.

mean a finite T_{0} space. For each A ⊂ X, the closure (resp. interior, exterior, boundary) of A will be denoted by \overline{A} (resp. int (A), ext (A), ∂A).

For each point x in a space X, there is a smallest neighborhood which is contained in each other neighborhood of x. For each x ∈ X, let

U_{x} = ⋂{V : V is an open set containing x}

Clearly U_{x} is the smallest open set containing x since X is finite.

Alexandroff spaces are the topological spaces in which each element is contained in a smallest open set or equivalently the spaces where arbitrary intersections of open sets are open. It is clear that all finite spaces are locally finite and all locally finite spaces are Alexandroff.

Lemma 1.1. The class U = {U_{x} : x ∈ X} is a base for a finite space (X, τ). Each base of τ contains U.

Notice that if X is Alexandroff, then X is T_{1} if and only if U_{x} = {x}. It follows that X is discrete and so every point is an isolated point.

Remark 1.2. Observe that if x and y are two points in a space X, then y ∈ U_{x} if and only if U_{y} ⊆ U_{x}.

Definition 1.3. [2] For two points x, y ∈ X, y ≥ x if U_{y} ⊆ U_{x}.

Remark 1.4. From Definition 1.3, the relation ≥ is reflexive and transitive since ⊆ is so.

Proposition 1.5. In a space X, y ≥ x if and only if x ∈ \overline{\{y\}}. In such case, x is said to be identified with y.

Proof. Let y ≥ x and y ≠ x. Then y ∈ U_{x} which is the smallest open set containing x. Then for any open set G containing x, we have (G \ {x}) ∩ \{y\} ≠ ϕ. This means x is an accumulation point of y. Therefore x ∈ \overline{\{y\}}. Conversely, let x ∈ \overline{\{y\}}. Then G ∩ \{y\} ≠ ϕ for every open sets G containing x. So y ∈ G for every open set G. Take G = U_{x}. By Remark 1.2, we get U_{y} ⊆ U_{x}. This shows that y ≥ x. □

The following example will be used throughout the paper.

Example 1.6 (The class of topologies on a set with three points). We give a list of all topologies on the set X = {x, y, z} up to homeomorphic topologies. There are 9 topologies, with the following U_{x}, U_{y} and U_{z}:

τ_{1} All three points are isolated. This is the discrete topology and U_{x} =

{x}, U_{y} = {y}, U_{z} = {z}. Now, we assume that the topology has only two isolated points , say, U_{x} = {x} and U_{y} = {y}. We have two cases concerning U_{z}:

τ_{2} The neighborhood of U_{z} has two points. We can assume U_{z} = {x, z}. τ_{3} The neighborhood of U_{z} has three points, U_{z} = X.

ON FINITE T_{0} TOPOLOGICAL SPACES 77

Next, we consider the case that the topology has only one isolated point, say, U_{x} = {x}. Again, we have to distinguish between different cases concerning U_{y} and U_{z}:

τ_{4} Both U_{y} and U_{z} have three points. Then U_{y} = U_{z} = X.

τ_{5} One neighborhood, say U_{z}, has three points and U_{y} has two points.

Then U_{y} = {y, z} is not possible, so U_{y} = {x, y}. τ_{6} Both neighborhoods have two points and are equal U_{y} = U_{z} = {y, z}. τ_{7} U_{x} and U_{y} have two points and are different. Then they can not be

{y, z}. Thus U_{y} = {x, y} and U_{z} = {x, z}. Finally, we have two cases without isolated points:

τ_{8} There is a neighborhood with two points, say, U_{x} = {x, y}. Then the neighborhood of z must have three points. Thus U_{x} = U_{y} = {x, y} and U_{z} = X.

τ_{9} All neighborhoods have three points, then U_{x} = U_{y} = U_{z} = X. Figure 1 shows this class of topologies with its minimal neighborhoods. The smallest ellipse refers to a singleton, the middle is a two points neighborhood and the biggest one is the whole space X.

2. T_{0} Properties and Associated Partial Order

In this section, we investigate some properties of T_{0} spaces such that for two distinct points x, y in X, x ≥ y and y ≥ x only true if U_{x} = U_{y}. If U(x) denote the neighborhood system of x. Recall that a space is T_{0} if and only if U(x) ≠ U(y) for x ≠ y.

Remark 2.1. The topologies τ_{i} for i ∈ {1, 2, 3, 5, 7}, in Example 1.6, are T_{0}. Any two elements in Figure 1 with the same neighborhood is not T_{0} such as τ_{j} for j ∈ {4, 6, 8, 9}.

Definition 2.2. For points x, y in a topological space X, y ≥ x if U(y) ⊇ U(x). Notice that ⊇ means more numbers of neighborhoods.

Remark 2.3. The relation ≥ is reflexive and transitive, since this is true for ⊇. In a T_{0} space, it is also antisymmetric. Then ≥ is a partial order in T_{0} spaces.

Definition 2.4. Let (X, ≤) be a partially ordered set and U ⊆ X. We say that U is open if whenever x ∈ U and y ≥ x it is also the case that y ∈ U.

In the following result we denote by the set of open subsets of X by U_{≤}.

Proposition 2.5. If (X, ≤) is a partially ordered set, then (X, U_{≤}) is a finite T_{0} space.

Proof. Clearly, X and ϕ are elements in U_{≤}. Let U_{i} ∈ U_{≤} for every i ∈ I. For any x ∈ ⋃_{i∈I} U_{i} and y ≥ x, there is i_{0} ∈ I such that x ∈ U_{i_{0}}. By openness

of U_{i_{0}}, we have y ∈ U_{i_{0}} and y ∈ ⋃_{i∈I} U_{i}. Therefore ⋃_{i∈I} U_{i} ∈ U_{≤}. Also, if A and

Figure 1. Topological spaces with minimal neighborhoods.

B are elements in U_{≤}, then A ∩ B ∈ U_{≤}. To show T_{0}, consider two distinct elements x, y ∈ X. Clearly, x ∈ U_{x} and y ∈ U_{y}. If y ∉ U_{x}, the proof is complete. If y ∈ U_{x}, y ≥ x, by the antisymmetry of ≤, x ≱ y and so x ∉ U_{y} which also completes the proof. □

Remark 2.6. The closure of any singleton p of a finite T_{0} space (X, U_{≤}) has the form \overline{\{p\}} = \{x : x ≤ p\}.

Remark 2.7. The order relation ≤ is not useful in T_{1} spaces, because there are no comparable elements. In other words, since for every two distinct points x and y in a T_{1} space, \overline{\{x\}} ≠ \overline{\{y\}}, then the relation x ≤ y is never satisfied. Thus ≤ is a good tool only for T_{0} spaces which are not T_{1}.

Proposition 2.8. A finite T_{0} space contains an isolated point.

Proof. Obvious. □

Remark 2.9. Finite spaces with isolated points need not be T_{0}. Each of τ_{4} and τ_{6} in figure 1 of example 1.6 has an isolated points, but not T_{0}.

Proposition 2.10. Every open set in a finite T_{0} space contains an isolated point.

Proof. Let G be an arbitrary open set in X. Then G is a finite open T_{0} subspace. By Proposition 2.8, there is an isolated point x in G. Since G is open in X, then x must be isolated point in X. □

3. Some Topological Properties

In this paper, some of topological operators which are well known for topologists have new forms with respect to the Alexandroff’s notion of order.

Proposition 3.1. In any finite space, the open (closed) points are the maximal (minimal) elements. Therefore U is an open set if it contains its upper bounds. By the complement, F is closed if it contains its lower bounds.

Proof. By Definition 1.3 and Proposition 1.5, the partial order x ≤ y for points x, y in a finite T_{0} space X was defined by y ∈ U_{x} or x ∈ \overline{\{y\}}. This means that for open points {x} = U_{x}, there is no strictly larger y. Thus the open points are the maximal elements in X. Similarly, for a closed point y, if U_{x} = {x}, then x ≤ y implies y = x. Hence the closed points are the minimal elements. □

Proposition 3.2. In a finite topological space X, the interior of a subset A ⊂ X, int A, is the set of all points a ∈ A such that for every b ≥ a it is also true that b ∈ A. In other words, the elements of int A are all upper bounds of X belonging to A.

Proof. We prove that int A = {a ∈ A : b ≥ a implies b ∈ A}. For “ ⊆′′, let a ∈ int(A) that is U_{a} ⊆ A for int A is the greatest open set contained in A. But U_{a} = {b : b ≥ a}. For “ ⊇′′, if b ∈ A for all b ≥ a. That means U_{a} ⊆ A, then a ∈ int A. □

Proposition 3.3. For a subset A in a finite topological space X, the closure of A, \overline{A}, is the set of points b ∈ X such that b ≤ a for some a ∈ A. In other words, \overline{A} is the set of all lower bounds of points of A.

Proof. In a finite topological space X, \overline{A} = ⋃_{a∈A} \overline{\{a\}}. By the definition of ≤,

we have \overline{\{a\}} = \{b : b ≤ a\} and the claim. □

Proposition 3.4. For a finite topological space X, the closure of an interior of a subset A of X, \overline{int A}, is the set of all x in X such that there exists a maximal element a ∈ A with a ≥ x.

Proof. By Proposition 3.3, \overline{int A} = ⋃\{b : b ≤ a for some a ∈ int A\}. Since a ∈ int A, then all c ≥ a are also in A. Therefore \overline{int A} = ⋃\{b : b ≤ a for a ∈ A}. Since we have a finite ordered set, there is a maximal c ≥ a. Hence \overline{int A} is the lower bounds of maximal elements in A. □

Proposition 3.5. In a finite topological space X, the interior of a closure of A ⊂ X, int \overline{A}, is the points x ∈ X such that all maximal elements y ≥ x belong to A.

Proof. From 3.2 and 3.3, we have

int \overline{A} = \{a ∈ \overline{A} : b ≥ a implies b ∈ \overline{A}\}
= \{a ∈ \overline{A} : b ≥ a implies that there is c ∈ A, c ≥ b\}
= \{a ∈ \overline{A} : all maximal elements b ≥ a must belong to A\}

To prove the equality, if b is maximal, then b ∈ A since there is no other c in A. Hence b ∈ int\overline{A}. On the other hand, if all elements in A are maximal and b ≥ a, then there is a maximal c ≥ b with c ∈ A. □

In finite topological spaces, From Propositions 3.4 and 3.5, we have int\overline{A} ⊆ \overline{int A}. The following example shows that the inverse inclusion is not always true.

Example 3.6. Consider the topology τ_{3} in Example 1.6. If A = {y, z}, we get int \overline{A} = \{y\} and \overline{int A} = \{y, z\}. Therefore int\overline{A} ⊉ \overline{int A}.

In 1961, Levine [16] introduced the notion of semi-open set in any topological space. A subset A in a topological space X is called semi-open if and only if A ⊂ \overline{int A}. In 1982, Mashhour et. al. [20] defined the concept of preopen set. A subset A is preopen in X if and only if A ⊂ int \overline{A}. There are no implications between these two concepts. This means that semiopen sets need not be preopen and conversely. In 1997, Abd El-Monsef et. al. [9] defined the concept of γ-open set as a union of semi-open and preopen sets which is equivalent to A ⊂ int \overline{A} ∪ \overline{int A}, for any subset A of X. From the relation int\overline{A} ⊆ \overline{int A}, we obtain a new implication between types of near openness in finite topological spaces.

Remark 3.7. In a finite topological space X, the following are true:

(i) Every preopen set in X is semi-open. The converse may not be true as it is shown in Example 3.6

(ii) Semiopen sets coincide with γ-open sets. The following diagram shows the relation between these notions in a finite case.

openness ⟹ preopenness
⇓
γ-openness ⟺ semi-openness

The axiom T_{0} is necessary for satisfying the previous implications. The following examples show this fact.

Example 3.8. Any subset of an indiscrete space is preopen but not semiopen, since the indiscrete spaces are not T_{0}.

Proposition 3.9. In a finite space X, the boundary of a subset A of X is the set b ∈ X such that b ≤ a for some a ∈ A and b ≤ c for some c ∉ A.

Proof. It follows from the definition, ∂A = \overline{A} ∩ \overline{X \ A}, and Proposition 3.3. □

Proposition 3.10. In a finite space X, the exterior of A ⊂ X is the set of points x ∈ X such that for b ≥ x implies b ∉ A.

Proof. Since for any topological space ext (A) = X \ \overline{A} = int (X \ A). By Proposition 3.2, we have

ext (A) = {x ∈ X : b ≥ x implies b ∈ X \ A}
= {x ∈ X : b ≥ x implies b ∉ A}
□

4. The Dimension for Finite Spaces

The notion of a topological dimension has a sense for finite spaces. Although we have only a finite number of points, all finite dimensions are possible. We use the notion of inductive dimension as stated by Engelking ([11], Chapter 7). A discrete space X has dimension zero denoted by dim X = 1, since the neighborhoods have no boundaries. On the real line R, the open intervals form a basis. The boundary of an interval consist of two points which is a discrete space. Analogously, the plane has dimension ≤ 2 since a circumference has dimension ≤ 1. This is the typical situation in three-dimensional spaces. This leads to the following definition.

Definition 4.1. ([7], p.195) Let X be a topological space. dim X = −1 if and only if X = ϕ. Let n be a positive integer and dim X ≤ k be defined for each k ≤ n−1. Then dim X ≤ n if X has a base β such that dim ∂B ≤ n−1 for all B ∈ β.

A base β of any finite space will be replaced by the set of all minimal neighborhoods for each of its points.

Proposition 4.2. For a finite space X, dim X ≤ n if and only if the minimal base U fulfils dim ∂U_{x} ≤ n − 1 for each U_{x} ∈ U.

Proof. If dim X ≤ n, then by definition 4.1 it has a base β such that dim ∂B ≤ n − 1 for all B ∈ β. By Lemma 1.1, each β contains U. Then dim ∂U_{x} ≤ n − 1. The converse is obvious. □

Proposition 4.3. A topological space X has dimension zero if and only if X has a base of clopen sets.

Proof. It is clear that ∂B = ϕ if and only if B is clopen. Dimension zero means existence of a base with ∂B = ϕ. This condition is equivalent to B being clopen. Since ∂B is containing those x which are accumulation points of B and its complement. For clopen sets B, we have ∂B = ϕ. On the other hand, if ∂B = ∂(X \ B) = ϕ, then B and X \ B are closed and so B is clopen. □

Proposition 4.4. Every finite T_{0} space with base β of clopen sets is discrete.

Proof. Since every zero dimensional T_{0} space is T_{1}. then it is closed as a consequence of the finiteness of the space. □

Finiteness and T_{0} axiom are necessary. It can be satisfied from the following examples.

Example 4.5. The Cantor set {0, 1}^{∞} with product topology has a base of clopen sets, but this space is not discrete.

Example 4.6. In Example 1.6, the topology τ_{6} has a base {{x}, {y, z}} with clopen sets. X is not discrete, because X is not T_{0}.

Lemma 4.7. Let X = C ∪ V such that every c ∈ C is closed and v ∈ V is open. Then each of C and V is a discrete subspace of X.

Proof. Since C is a finite closed subspace of X, then each c ∈ C is a closed point in C. Then C\{c} is a finite closed subset in C. So {c} is an open point in C. This means that C is a discrete set. It is clear that V is discrete. □

The following Proposition describes the finite one-dimensional T_{0} space.

Theorem 4.8. Let X be a finite T_{0} space. Then dim X ≤ 1 if and only if every singleton in X is either open or closed.

Proof. Let X be a finite T_{0} space. By Proposition 2.8, X has an open point say x_{0} and U_{x_{0}} = {x_{0}}. Since dim X ≤ 1, then dim ∂({x_{0}}) = 0 and so ∂({x_{0}}) is discrete. This means that each y_{0} ∈ ∂({x_{0}}) is closed in ∂({x_{0}}). Since ∂({x_{0}}) is closed in X, then {y_{0}} is also closed in X. Take X′ = X \ cl({x_{0}}) which is an open finite T_{0} subspace of X. By Proposition 2.10, X′ has an open point set x_{1} and U_{x_{1}} = {x_{1}} which is also open in X. Also dim ∂({x_{1}}) = 0, then ∂({x_{1}}) is discrete. So, each y_{1} ∈ ∂({x_{1}}) is closed in ∂({x_{1}}). Then {y_{1}} is closed in X. Take X′′ = X \ cl({x_{0}, x_{1}}). By continuing, the proof is completed in one direction. Conversely, Suppose that singletons are open or closed. By Lemma 4.7, we have in either cases discrete subspaces of X. Therefore the dimension of each subspace is X ≤ 1. Hence dim X ≤ 1. □

Definition 4.9. The height of a partially ordered set (X, ≤) is the degree of a longest element of the increasing sequence x_{1} < x_{2} < .... < x_{n} of elements of X.

Example 4.10. The discrete space has a height 1, since there are no comparable elements i.e. no x < y. The space in which each point is either open or closed of height 2, for x < y is satisfied only for a closed point x and an open point y.

Theorem 4.8 can be reformulate as follows: The dimension of a finite space X is the height of a partially ordered set (X, ≤) minus 1.

Remark 4.11. Recall that a space (X, τ) is called T_{\frac{1}{2}} if every generalized closed subset of X is closed [18] or equivalently if every singleton is either open or closed [8]. Observe that by Theorem 4.8, every finite T_{\frac{1}{2}} space is a one dimensional T_{0}. It is also clear that a space X is T_{\frac{1}{2}} if its non-closed singletons are isolated.

5. Density in Finite Spaces

In this section, we introduce some characterizations of density in finite spaces.

Proposition 5.1. In finite T_{0} spaces, the set of all open points is dense.

Proof. Let X be a finite T_{0}. By Proposition 2.8, X contains an isolated point. If O = {x : {x} is open }. Then by Proposition 2.10, every open set G containing x must intersect O. This means that x ∈ \overline{O}. Hence O is dense. □

Remark 5.2. The condition T_{0} is necessary in Proposition 5.1. The singleton {x} in τ_{6} of example 1.6 is open but not dense.

Proposition 5.3. A subset A is dense in a finite T_{0} space X if and only if it contains all open points of X.

Proof. Let O = {x : {x} = U_{x}} be the set of open points in X. If A is a dense subset of X, by density, then A must intersect U_{x}. So, it must contain x. The converse case follows readily from Proposition 5.1. □

Corollary 5.4. In finite T_{0} spaces, each dense set contains an open dense set.

Proof. Obvious from Proposition 5.1 and Proposition 5.3. □

In Corollary 5.4, The condition T_{0} is necessary.

Example 5.5. Consider the topology τ_{6} in Example 1.6. A subset {a, c} is dense. But it does not contain an open dense subset.

Some topological concepts induced by density. Recall that A is said to be codense (resp. nowhere dense, dense-in-itself) if int (A) = ϕ (resp. int \overline{A} = ϕ, A = d(A)), where d(A) denotes the set of accumulation points of A. Dense-in-itself of any A ⊆ X, equivalently that A does not have any isolated points. We state these notions in the finite spaces.

Proposition 5.6. For arbitrary finite topological space X, the following are true for any subset A of X,

(i) It is co-dense if there is no upper bounds belongs to A.

(ii) It is nowhere dense if there is no element has a maximal element in

A. (iii) It is dense in-itself if A contains all of its lower bounds.

Proof. (i) and (ii) are obvious from Proposition 3.2 and Proposition 3.5 respectively. (iii) Clear by Proposition 3.3 and using the equality A = \overline{A}. □

Recall that a topological space (X, τ) is said to be a submaximal if each of its dense subsets is open.

Proposition 5.7. Any finite T_{0} space X is submaximal if and only if it contains at most a non-isolated point.

Proof. The set of open points is dense by Proposition 5.1. Also by the submaximality of X, every dense subset is open. Then for every points x, y, z ∈ X such that x ≤ y ≤ z, X \ {y} is not open. Therefore y is not isolated. Conversely, if all the points of X are isolated, then the only dense set is X itself. This means that X is submaximal. Otherwise, if X contains only non-isolated point y. Let A be any dense subset of X. By Proposition 5.3, A contains all isolated points of X. □

Corollary 5.8. Every finite T_{0} submaximal spaces is T_{\frac{1}{2}} . It is also onedimensional space.

Proof. It is a consequence Theorem 4.8. □

6. Some Weaker Forms of Continuity in Finite Spaces

Continuous functions play an important role in topology. The following proposition shows a corresponding definition of continuity between finite spaces using the minimal neighborhood of each point and its image.

Proposition 6.1. Let X and Y be finite topological spaces, then the function f : X ⟶ Y is continuous at x if and only if f(U_{x}) ⊆ U_{f(x)}.

Proof. Let f : X ⟶ Y be a continuous function. Fix a point x ∈ X. Since U_{f(x)} is an open neighborhood of f(x), then by the continuity of f, f^{−1}(U_{f(x)}) is an open neighborhood of x and so f(U_{x}) ⊆ U_{f(x)}. Conversely, let W be an open set containing f(x). By assumption, f(U_{x}) ⊆ U_{f(x)} ⊆ W. Take U_{x} = U, then f(U) ⊆ W which shows that f is continuous at x. □

Recall that a function f : X ⟶ Y from a topological space X into a topological space Y is precontinuous [20] (resp. semicontinuous [17], γcontinuous [9]) if the inverse image of each open set in Y is preopen (resp. semiopen, γ-open) in X.

In an arbitrary topological space, there is no connection between precontinuity and semicontinuity. Each of them implies γ-continuity. In the finite case, by Remark 3.7, every precontinuous function is semicontinuous which coincides with γ-continuous. The following implications show a new connection between continuity and some kinds of near continuity.

Continuity ⟹ Precontinuity
⇓
γ-continuity ⟺ Semicontinuity

The following example shows that the converse is not always true.

Example 6.2. Let X = Y = {x, y, z} as in Example 1.6. The mapping f : (X, τ_{3}) ⟶ (Y, τ_{5}) which defined by f(x) = x, f(y) = z and f(z) = y is γ-continuous, but not precontinuous.

A function f : X ⟶ Y from a topological space X into a topological space Y is called preopen [20] (resp. semiopen[6], γ-open[9]) if the image of each open set in X is preopen (resp. semiopen, γ-open).

In arbitrary topological space, there is no connection between preopen and semiopen functions. In a finite case, by Remark 3.7, a preopen function is semiopen which coincides with γ-open. Each of them belongs to γ-open function. The implications between these types of functions and other corresponding ones are given by the following diagram:

Open function ⟹ Preopen function
⇓
γ-open function ⟺ Semiopen function

The converse of these implications are not true, in general, as the following example illustrates.

Example 6.3. Let X = {x, y, z} with a topology τ_{5} as in Example 1.6 and Y = {a, b, c, d} with the topology of minimal neighborhoods U_{a} = {a}, U_{b} = {b}, U_{c} = U_{d} = Y. The function f : X ⟶ Y is γ-open but not preopen.

7. Continuity of Multifunctions in Finite Topological Spaces

Let X and Y be two nonempty sets and P (Y) be the collection of all subsets of Y. Recall that the function F : X ⟶ P (Y) is called a multifunction ([7], p.186).

Definition 7.1. A function F : X ⟶ P (Y) from a topological space X into a topological space Y is called:

(1) upper semicontinuous mulitifunction (abb.U.S.C) at a point x_{0} ∈ X if for every open set V in Y such that F (x_{0}) ⊂ V, there exists an open set U containing x_{0} such that F (U) ⊂ V.

(2) lower semicontinuous multifunction (abb.L.S.C)at a point x_{0} ∈ X if for every open set V in Y such that F (x_{0}) ∩ V ≠ ϕ, there exists an open set containing x_{0} such that F (x) ∩ V ≠ ϕ for each x ∈ U.

Remark 7.2. U.S.C is not necessarily L.S.C, and L.S.C is not necessarily U.S.C.

Now, the following theorems and examples give the relation beteen the usual definition of continuity for single valued functions and upper and lower semicontinuity for multifunctions.

Definition 7.3. [14] A space X is said to be a connected ordered topological space (abb. COTS) if for every three points subset Y in X, there exists y ∈ Y such that Y meets two connected components of X \ {y}. In other words, for any three points one of them separates the other two.

Remark 7.4. Let X = [0, 1] with usual topology and f be a continuous function from X into itself. π : X ⟶ Y is a quotient function from X into a finite COTS Y. The function g from Y into itself is defined by g(y) = πfπ^{−1}{y} is continuous. The following examples discuss the continuity of g in the case of single valued functions.

Example 7.5. Let X = [0, 1] and f : X ⟶ X be a continuous function defined by

f(x) = \{ \begin{matrix}x + 1/2 : & 0 ≤ x ≤ 1/2 \\ 1 : & 1/2 ≤ x ≤ 1 \end{matrix}

and Y = {a, b, c, d, e} is a COTS. π is a quotient function defined by

π(x) = \{ \begin{matrix}a : & x = 0 \\ b : & 0 < x < 1/2 \\ c : & x = 1/2 \\ d : & 1/2 < x < 1 \\ e : & x = 1 \end{matrix}

Since g(y) = πfπ^{−1}(y) for each y ∈ Y, so g(a) = c, g(b) = d, g(c) = e, g(d) = e, g(e) = e. Then g is continuous at each point y ∈ Y.

The following theorem is considered as an equivalent definition for U.S.C and L.S.C in finite topological spaces with respect to the minimal neighborhood of each point.

Theorem 7.6. For finite topological spaces X and Y and multifunction F : X ⟶ P (Y), we have:

(i) F is U.S.C at x if and only if F (U_{x}) ⊂ ⋃_{y∈F (x)} U_{y}.

(ii) F is L.S.C at x if and only if F (x′) ∩ U_{y} ≠ ϕ for all x′ ∈ U_{x} and y ∈ F (x).

Proof. (i). Let F be U.S.C at x ∈ X. Then for every open set V such that F (x) ⊂ V. There exits an open set U with x ∈ U such that F (U) ⊂ V. Since x ∈ U, then U_{x} ⊂ U. By assumption, we obtain F (U_{x}) ⊂ V. Take

V = ⋃_{y∈F (x)} U_{y}, then F (U_{x}) ⊂ ⋃_{y∈F (x)} U_{y}. Conversely, let F (U_{x}) ⊂ ⋃_{y∈F (x)} U_{y}.

For x ∈ U_{x}, take any V ⊃ F (x). As U, we choose U_{x}, then F (U_{x}) ⊂ ⋃_{y∈F (x)} U_{y} ⊂ V. Therefore F (U_{x}) ⊂ V shows that F is U.S.C.

(ii). Let F be L.S.C at x ∈ X. That means for every open set V such that F (x) ∩ V ≠ ϕ, there is an open set U with x ∈ U such that F (x′) ∩ V ≠ ϕ for all x′ ∈ U. Applying for a special V, take V = U_{y} and y ∈ F (x). Since U_{x} ⊂ U, then F (x′) ∩ U_{y} ≠ ϕ for all x′ ∈ U_{x} and y ∈ F (x). Conversely, let V be an arbitrary open set with F (x) ∩ V ≠ ϕ. Take y ∈ F (x) ∩ V, then Uy ⊂ V. As U, we choose U_{x} and by assumption ϕ ≠ F (x′)∩U_{y} ⊂ F (x′)∩V. Then F (x′) ∩ V ≠ ϕ for all x′ ∈ U_{x} and F is L.S.C. □

U.S.C may not be L.S.C. It can be satisfied from the following example.

Example 7.7. Let X = Y = {a, b}. Their minimal neighborhoods are U_{a} = {a} and U_{b} = {a, b}. The multifunction F : X ⟶ P (Y) is defined by F (a) = {a, b}, F (b) = {a}. Then F is L.S.C. It is not U.S.C since we have

F (U_{b}) = {a, b} ⊄ ⋃_{y∈F (b)} U_{y} = {a} by Theorem 7.6(i), U_{b} = {a, b}, F (b) =

{a}. Therefore F is not U.S.C. If the multifunction G : X ⟶ P (Y) is defined by G(a) = {a}, G(b) = {a, b}, then G is U.S.C. It is not L.S.C at the point b. Since by Theorem 7.6(ii), we have U_{b} = {a, b}, F (b) = {a, b} and F (a) ∩ U_{a} = {b} ∩ {a} = ϕ.

The following examples discuss the upper and lower semi-continuity of g.

Example 7.8. Let X = [0, 1] with usual topology. The function f from X into itself defined by

f(x) = \{ \begin{matrix}3/4 − 2x : & 0 ≤ x < 1/4 \\ 1/3x + 1/6 : & 1/4 ≤ x ≤ 1 \end{matrix}

and the quotient function π : X ⟶ Y, where Y is a finite COTS with five points defined by

π(x) = \{ \begin{matrix}a : & x = 0 \\ b : & x = 0 < x < 1/2 \\ c : & x = 1/2 \\ d : & 1/2 < x < 1 \\ e : & x = 1 \end{matrix}

The multifunction g : Y ⟶ P (Y) is defined by g(y) = πfπ^{−1}y. Then g(a) = d, g(b) = {b, c, d}, g(c) = b, g(d) = b, g(e) = c. g is L.S.C, but not U.S.C since the point y = a, U_{a} = {a, b}, g(a) = {d} and g(U_{a}) = {b, c, d} ⊄

⋃_{y∈g(a)} U_{y}) = {d}.

Now, we consider some properties of U.S.C and L.S.C for multifunctions. These make a connection between multifunctions and dimension theory.

Theorem 7.9. Let π : X ⟶ Y be a continuous function from a topological space X into a one-dimensional T_{0} space Y. For any closed point in Y, π^{−1}(y) is only one point and for any open point z ∈ U_{y}, π^{−1}(y) ∈ cl(π^{−1}(z)), then π is an open function.

Proof. Let W be an open set in X. Our aim is to prove that π(W) = ⋃_{y∈π(W)} U_{y}. It is clear that π(W) ⊂ ⋃_{y∈π(W)} U_{y}. So, it is enough to prove

that U_{y} ⊂ π(W) for each y ∈ π(W). There are two cases: If y is an open point, then U_{y} = {y} ⊂ π(W) and we are done. If y is a closed point, by the assumption, π^{−1}(y) = x. Let z be an open point such that z ∈ U_{(}y), by hypothesis, x ∈ cl(π^{−1}(z)). Since W is an open neighborhood of x, then π^{−1}(z)∩W ≠ ϕ. For a point v ∈ π^{−1}(z)∩W ≠ ϕ, we have z = π(v) ∈ π(W). Therefore, U_{y} ⊂ π(W). This completes the proof. □

Under the previous assumptions of Theorem 7.9, π is not a closed function.

Example 7.10. Let X = [0, 1] with usual topology and Y is a COTS with three points. The function π : X ⟶ Y is defined by

π(x) = \{ \begin{matrix}a : & x = 0 \\ b : & 0 < x < 1 \\ c : & x = 1 \end{matrix}

The assumption of Theorem 7.9 hold. π is not closed because if we take a closed set F ⊂ (0, 1), then π(F) = {b} is not closed.

Proposition 7.11. For a topological space X and one-dimensional T_{0}-space Y, consider a continuous function f and a quotient a function π from X into Y. If the multifunction g : Y ⟶ P (Y) is defined by g = πfπ^{−1}, then g(y) is a singleton for each closed point y ∈ Y.

Proof. Obvious, since f and π are single -valued functions. □

Theorem 7.12. For a space X and one-dimensional T_{0}-space Y, assume the following assumptions hold.

(1) The function π : X ⟶ Y is continuous.

(2) For every closed point y, π^{−1}(y) is one point.

(3) For every open point z ∈ U_{y}, π^{−1}(z) = x.

(4) A multifunction g : Y ⟶ P (Y) defined by g = πfπ^{−1}.

Then g is L.S.C.

Proof. Let y_{0} ∈ Y and V be an open set in Y such that g(y_{0}) ∩ V ≠ ϕ. If y_{0} is an open point, then we can take U_{y_{0}} = {y_{0}} and so g is L.S.C. If y_{0} is a closed point, by assumption, for any open point y ∈ U_{y_{0}}, π^{−1}(y_{0}) = x ∈ cl(π^{−1}(y)). By Proposition 30, g(y_{0}) is a singleton and so f^{−1}π^{−1}(V) is an open neighborhood of x. Therefore π−1(V) ∩ W ≠ ϕ for all y ∈ U_{y_{0}} and W = f^{−1}π^{−1}(V). This shows that g is L.S.C. □

Under the conditions of Theorem 7.12, the multifunction is not necessarily to be U.S.C.

Example 7.13. Let X and Y and the quotient function as in Example 7.8. The function f is defined by

f(x) = \{ \begin{matrix}2x − 1 : & 0 < x < 1/2 \\ 1 − 2x : & 1/2 < x < 1 \\ 1 : & x ∈ \{0, 1\} \end{matrix}

We have g(a) = e, g(b) = {b, c, d}, g(c) = a, g(d) = {b, c, d}, g(e) = e. g is not U.S.C since at the point y_{0} = a, U_{a} = {a, b}, g(a) = {e}, we have

g(U_{a}) = {b, c, d, e} ⊄ ⋃_{y∈g(a)} U_{y}) = {d, e}.

Acknowledgment. The results of this paper were obtained during my Ph.D. studies at Tanta University and are also contained in my thesis.^{1} I would like to express my deep gratitude to my supervisors whose guidance and support were crucial for the successful completion of this paper.

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Ernst-Moritz-Arndt-Universität, Institut für Mathematik und Informatik, Jahnstr. 15a, 17487 Greifswald, Germany

   E-mail address: elatik@mail.uni-greifswald.de

From November 20,2001 the new address will be, Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt

   E-mail address: aatik@dec1.tanta.eun.eg

Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt

Department of Mathematics, Faculty of Engineering, Tanta University, Tanta, Egypt