FINITE TOPOLOGICAL SPACES AND QUASI-UNIFORM STRUCTURES

P. Fletcher

(received March 28, 1969)

1. Introduction. In [6], H. Sharp gives a matrix characterization of each on a finite set X = {x , x , . . . , x } . The study of quasi-uniform spaces provides a more natural and obviously equivalent characterization of finite topological spaces. With this alternate characterization, results of quasi-uniform theory can be used to obtain simple proofs of some of the major theorems of [1], [3] and [6]. More­ over, the class of finite topological spaces has a quasi-uniform property which is of interest in its own right. All facts concerning quasi-uniform spaces which are used in this paper can be found in [4].

2. Preliminaries.

DEFINITION. Let X be a non-empty set and let U be a filter base on X X X such that

i) each element of \L is a reflexive relation on X ;

ii) if U e U there exists W e U such that W o W C U .

Then li is a quasi-uniformity on X . If U is a filter, then *Vl is called a quasi-uniform structure.

DEFINITION. Let X be a set and let U be a quasi-uniformity on X . Let 3 = {A C X : if a e A then there exists U e U such that U(a) C A} . Then -T is the quasi-uniform topology on X generated by UL.

DEFINITION. Let (X , cT) be a and let ÎJL be a quasi-uniformity on X. Then IX is compatible if rr = ^11 •

It is shown in [5] that if (X , 3") is a topological space, then there exists a compatible quasi-uniformity U on X.

3. Finite topological spaces. It is clear that every finite topological space has a finite compatible quasi-uniformity. Let (X , IT) be a topological

Canad. Math. Bull. vol. 12, no. 6, 1969

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Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 06:08:04, subject to the Cambridge Core terms of use. space with a finite compatible quasi-uniformity \L' and let |jL be the quasi-uniform structure generated by U1. Let U = O {V : V eU} . Then U e U . Thus there exists W G U such that W ° W C U . But UCU°UCW°WCU. Thus U is a reflexive and transitive relation, and since U is exactly the collection of all supersets of U, {U} is a compatible qua si-uniformity.

DEFINITION. Let (X, 3") be a topological space with a finite compatible qua si-uniformity It. Let U = O {V : V G \JL} . Then {U} is called the fundamental quasi-uniformity of (X , U') with respect to U-

THEOREM 3.1. Let (X , IT) be a topological space with a finite compatible quasi-uniformity and let XL = {U} be a compatible fundamental quasi-uniformity. Then for each x G X, U(x) is the smallest open set which contains x .

Proof. Let x G X . For any quasi- it is true that x G [U(x)]°. Hence U(x) C [U(x)]° . Moreover, if x e A e Î = 3" , then U(x) G A by definition.

THEOREM 3.2. Let (X , 3) be a topological space with a finite compatible quasi-uniformity and let It = {U} be a compatible fundamental quasi-uniformity. Let x,y G X with x £ y. Then (x, y) G U if arid only if U(y) C U(x) .

Proof. Suppose (x,y) G U. Then y G U(X) SO that U(y) C U ° U(x) = U(x) . The reverse implication follows from the fact that y G U(y) .

COROLLARY. _Let x,y G X with x # y. Then U(x) = U(y) if and only if y G U(X) and x G U(y) .

THEOREM 3.3. Let (X , 3") be a topological space and suppose there exists a finite compatible quasi-uniformity W. ' . Then the quasi- uniform structure li. generated by \i ' is the largest compatible quasi- uniform structure on X .

Proof. Let V be the universal quasi-uniform structure for (X , 3). By definition U C Y, and *Y is the largest compatible quasi-uniform structure on X. Let V G V and let U = D {W : W G U '} . Let x G X. Since U(x) is a subset of any open set which contains x, U(x) C [V(x)]° C V(x) . Thus U C V so that V G U. Hence It = *Y -

COROLLARY. Let (X , 7) be a finite topological space. Then there exists exactly one quasi-uniform structure which is compatible with (X,S) .

COROLLARY. Each topological space with a finite compatible quasi-uniformity has exactly one compatible fundamental quasi-uniformity.

772

Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 06:08:04, subject to the Cambridge Core terms of use. COROLLARY [6]. Let X be a finite set. There is a one-to-one correspondence between the collection of all on X and the collection of all reflexive, transitive relations on X .

COROLLARY [3]. The number of topologies on a finite set X / _ A \ with exactly n elements is less than or equal to 2

/ _ A \ Proof. There are 2 subsets of X X X - A . Hence there

are at most 2 fundamental quasi-uniformities on X .

4. Topological properties of finite topological spaces.

THEOREM 4.1. Let (X , 3) be a finite topological space and let It = {U } be the compatible fundamental quasi-uniformity. Then (X , 3" ) is connected if and only if {X,^ } is connected. U l UL l

Proof. Let aeX. Then by [4, Theorem 1.15], _1 -1 {a} = Pi {U(a) : U e U} = U (a). Thus every 3* _ j - open set is 3" - closed and the theorem follows. U

COROLLARY [6]. Jf X is a finite set there are an even number of non-trivial connected topologies on X .

The following theorem originally proved in [1] illustrates the force of quasi-uniform theory on finite topological spaces. All of the equivalences of this theorem are immediate consequences of basic theorems about quasi-uniform spaces.

THEOREM 4.2. [1]. Let (X, 3") be a finite topological space and let *U, = {U } be the compatible fundamental quasi-uniformity. The following are equivalent.

(a) U is symmetric;

(b) (X^) is regular;

(c) (X , 3*) is completely regular;

(d) (X , 3") jis. 0-dimensional;

(e) . (X,3-) is_ ft

(f) (X.3-) is RQ.

Proof. It is well known that a => c, c => d, and e => f .

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Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 06:08:04, subject to the Cambridge Core terms of use. b => a: [4, Theorem 3.17 ii].

a =» e: U(a) = U_1(a) = n{U_1(a) : U eU} = {T} .

f => a: [4, Theorem 3.8].

d => b: If U(a) is closed, then by [4, Theorem 1. 15], U(a) = U ° U(a) . By [4, Theorem 3. 17 iii], (X , 3") is regular.

Perhaps the most important separation axiom in finite topological spaces is the T separation axiom. By [4, Theorem 3.1], a finite topological space (X ,3) is T if and only if the member of its fundamental quasi-uniformity is anti-symmetric.

5. A quasi-uniform property of finite topological spaces.

Finite topological spaces have a noteworthy quasi-uniform property: namely, if (X , 3) is a finite topological space then (X , 7) has a unique compatible quasi-uniform structure. Uniform spaces having a unique compatible uniform structure have been characterized by R. Doss [Z], and it is well known that the unique compatible uniform structure of a compact is the collection of all neighborhoods of the diagonal in XXX. On the other hand, it is not difficult to find compact Hausdorff spaces which have two distinct compatible quasi-uniform structures. Such an example is given in [5], I conjecture that a topological space (X , 3) admits exactly one compatible quasi-uniform structure if and only if 3 is finite. The following theorem supports this conjecture.

THEOREM 5.1. Jf (XJ) is a with a unique compatible quasi-uniform structure, then (X , 3) is a finite topological space.

Proof. The unique quasi-uniform structure U. is the Pervin quasi-uniform structure. Since 3 = 3" is T , 3 -i is the discrete \jL l 1 \L x topology and since (X , 3") is uniformizable 3 = 3* . Thus (X , 3". ) is discrete and hence metrizable. It follows that XI» is complete and since XH is the Pervin quasi-uniform structure, OX is totally bounded. Then (X, 3* ) is compact and discrete. Hence X is finite.

The author wishes to thank the referee for his many helpful suggestions.

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Downloaded from https://www.cambridge.org/core. 25 Sep 2021 at 06:08:04, subject to the Cambridge Core terms of use. REFERENCES

1. D.A. Bonnet and J.R. Porter, Matrix characterizations of topological properties. Canad. Math. Bull. 11 (1968) 95-105.

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3. V. Krishnamurthy, On the number of topologies on a finite set. Amer. Math. Monthly 73 (1966) 154-157.

4. M.G. Murdeshwar and S.A. Naimpally, Qua si-uniform topological spaces. (Noordhoff, 1966.)

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Virginia Polytechnic Institute Blacksburg Virginia 24061

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