Some Properties of Semi-Continuous Functions and Quasi-Uniform Spaces

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MATEMATIQKI VESNIK UDC 515.123.2

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47 (1995), 11{18

research pap er

SOME PROPERTIES OF SEMI-CONTINUOUS

FUNCTIONS AND QUASI-UNIFORM SPACES

J. Ferrer, V. Gregori and I. L. Reilly

Abstract. This pap er approaches the problem of characterizing the top ological spaces that

admit a unique compatible quasi-uniform structure by means of semi-continuous functions. The

desired result is achieved by relating prop erties of the semi-continuous quasi-uniformitywitha

new characterization of hereditarily compact spaces via lower semi-continuous functions.

1. Intro duction

In the following we establish the basic de nitions, general results and termi-

nology to b e used throughout this pap er; they are found mostly in [2].

By a quasi-uniform space (X; U ) we mean the natural analog of a uniform

space obtained by dropping the symmetry axiom. For each x 2 X and U 2 U ,

U (x)=f y 2 X :(x; y ) 2 U g de nes a basic neighborhood of x for a top ology in

X which is the top ology induced by U . Considering the collection U of the sets

U = f (x; y ):(y; x) 2 U g, U 2U,we obtain the conjugate quasi uniform space

1 1

(X; U ). The collection of sets of the form U \ U , U 2U, is a subbase for a

uniform structure (X; U ), the uniform space induced by(X; U ).

A quasi-uniform space (X; U ) is precompact (totally b ounded) provided that

for each U 2 U there is a nite subset F of X ( nite cover C of X ) such that

U (F )=X (A A is contained in U , for each A 2C). Although total b oundedness

implies precompactness, the converse is not true for quasi-uniform spaces in general.

In any top ological space X , for eachopensetG,thesetS (G)=f (x; y ):x 2= G,

or x; y 2 G g de nes a subbasic entourage of a compatible transitive totally b ounded

quasi-uniformity P usually referred to as Pervin's quasi-uniformity [9], whichisthe

nest of all totally b ounded compatible quasi-uniformities in X [2, Prop. 2.2].

By dropping the symmetry axiom of a pseudometric (metric) one obtains a

quasi-pseudometric (quasi-metric). The euclidean uniform structure E of the real

AMS Subject Classi cation : 54E 35; 54E 55

Keywords : Semi-continuous function, quasi-uniform space, hereditarily compact space

The rsttwo authors have b een partially supp orted by a grant from the DGICYT, No.PB89-

0611, and the third author by a grant from the UniversityofAuckland Research Committee. 11

12 J. Ferrer, V. Gregori, I. L. Reilly

line R may b e decomp osed into two conjugate quasi-pseudometric spaces (R; Q),

(R; Q ), where Q denotes the quasi-uniformity induced by the quasi-pseudometric

d(x; y )= max(x y; 0), i.e., the sets V = f (x; y ):x y0, form a base

for Q and Q = E .

For a function b etween two quasi-uniform spaces, quasi-uniform continuityis

de ned in the analogous way to the uniform case.

The symb ols LS C (X ), USC(X ), LS C B (X ), USCB(X ), LS C B B (X ),

USCBA(X ) will indicate the collections of all real valued functions whicharelow-

er semi-continuous, upp er semi-continuous, lower semi-continuous b ounded, upp er

semi-continuous b ounded, lower semi-continuous b ounded below and upp er semi-

continuous b ounded ab ove, resp ectively.

Given a quasi-uniform space (X; U ), the symbols Q(U ), Q (U ), E (U ),

[QB (U ), Q B (U ), EB(U )] will stand for the collections of all real valued functons

in X which are quasi-uniformly continuous [and b ounded] from (X; U )to(R; Q),

(R; Q ) and (R; E ), resp ectively.

It is well known that the ring of continuous real valued functions C (X ) on

a completely regular space determines a compatible uniformity C , which is the

coarsest for which every element of C (X ) is uniformly continuous. In the same

way, given any top ological space X , for each f 2 LS C (X ) and r > 0, the set

U (f; r)=f (x; y ):f (x) f (y )

quasi-uniformity SC which is the coarsest of all compatible quasi-uniformities U for

which LS C (X )iscontained in Q(U ), see [2, Prop. 2.10].

The purp ose of this pap er is to relate semi-continuity to quasi-uniform con-

tinuity by means of the quasi-uniformity SC. This relationship will turn out to

b e useful as it will provide a characterization of those top ological spaces that ad-

mit a unique compatible quasi-uniform structure. Lindgren [8] has shown that a

top ological space admits a unique compatible quasi-uniformity if and only if every

compatible quasi-uniformity is totally b ounded.

No separation prop erties are assumed for the spaces used throughout this pap er

unless explicitly stated.

2. Semi-continuity and quasi-uniform continuity

In this section we compare the notions of semi-continuous and quasi-uniformly

continuous functions with resp ect to the structures Q, Q and E , and obtain the

interesting equality E (P ) = C (X ), i.e., the b ounded continuous functions in X

coincide with the functions that are quasi-uniformly continuous with resp ect to

Pervin's quasi-uniformity. We also show that the quasi-uniform space (X; SC) is

totally b ounded if and only if every semi-continuous function in X is b ounded.

2.1. Proposition. If (X; U ) is a precompact quasi-uniform space, then the

fol lowing inclusion relations hold

Q(U ) LS C B B (X ); Q (U ) USCBA(X ); E (U ) C (X ):

Some prop erties of semi-continuous functions . . . 13

Proof. Weshow only the rst inclusion since the second is analogous and the

third one follows from the rst two and the fact that E (U )=Q(U ) \ Q (U ).

Let f 2 Q(U ). Since f is continuous from X to (R; Q), wehave f 2 LS C (X ).

By quasi-uniform continuity, there is U 2 U such that f (x) f (y ) < 1, for all

(x; y ) 2 U . Since (X; U ) is precompact, there is a nite subset F of X suchthat

X = U (F ). Let r = minf f (x) 1 : x 2 F g, then f (x) > r , for all x. Thus,

f 2 LS C B B (X ).

2.2 Proposition. For any topological space X the fol lowing inclusion rela-

tions hold

LS C B (X ) Q(P ); USCB(X ) Q (P ):

Proof. Again, weprove only the rst inclusion. Let f be in LS C B (X ). For

each r>0, since f is b ounded, there is a nite subset F of R suchthatf (X )is

contained in the nite union of op en intervals f (t r=2;t + r=2) : t 2 F g.

For each t 2 F , set G(t)=f x 2 X : f (x) >t r=2 g. Then G(t) isanopen

subset of X and the entourage U = f S (G(t)) : t 2 F g is in P .

Now, for (x; y ) 2 U ,lett 2 F be suchthat t r=2

Then x 2 G(t)andy 2 G(t). Thus f (x) f (y )=f (x) t + t f (y )

f 2 Q(P ).

2.3 Corollary. In any topological space X the fol lowing relations hold

LS C B (X )= QB (P ) Q(P ) LS C B B (X ),

1 1

USCB(X )= Q B (P ) Q (P ) USCBA(X ),

EB(P )=E (P )=C (X ).

2.4 Proposition. In a topological space X , the semi-continuous quasi-

uniformity SC is total ly bounded if and only if every semi-continuous function is

bounded.

Proof. For the necessity part, it suces to showthat LS C (X )=LS C B (X ).

Let f 2 LS C (X ). There is a nite cover A ;A ; ...;A of X suchthat A A

1 2 n i i

U (f; 1), 1 i n. For each i =1; 2; ...;n,cho ose x 2 A . Then,

i i

1 1

A U (f; 1)(x ) \ U (f; 1) (x )=f (f (x ) 1;f(x )+1):

i i i i i

Thus, f (A ) is b ounded for 1 i n, and so is f (X ).

Conversely, from Prop osition 2:2, LS C (X ) = LS C B (X ) Q(P ). But the

construction of SC implies that it is equal to P . Since P is totally b ounded, so is

SC.

14 J. Ferrer, V. Gregori, I. L. Reilly

3. Semi-continuous functions and unique quasi-uniformity spaces

The problem of nding a purely top ological characterization of the spaces

that admit only one compatible quasi-uniformity was rst posed by Fletcher [1]

in 1969. Afewyears ago, Kunzi [6] came up with the desired characterization by

showing that a top ological space admits a unique quasi-uniformity if and only if it

is hereditarily compact (every subset is compact) and it has no strictly decreasing

sequence of op en sets with op en intersection.

Our aim here is to achieve a similar result to that of Kunzi in terms of lower

semi-continuous functions.

Hereditarily compact spaces were studied intensively by Stone [10]. He char-

acterized them in several di erentways, although not in terms of semi-continuous

functions. Next, we give a characterization of hereditariy compact spaces using

semi-continuous functions which will b e needed later.

3.1. Proposition. Atopological space X is hereditarily compact if and only

if the range f (X ) of every lower semi-continuous function f is wel l orderedinR.

Proof. Let f 2 LS C (X ). If f (X ) were not well ordered in R, we could

construct a strictly decreasing sequence (f (x )). For each n, let G = f x 2 X :

n n

f (x) > f (x ) g. Then (G ) is a strictly increasing sequence of op en sets, which

n n

contradicts the fact that every subset is compact.

Conversely,if X were not hereditarily compact, we could nd a strictly increas-

ing sequence of op en sets (G ). For each n, let g be the characteristic function of

n n

G . The function

f =sup

is then lower semi-continuous, but f (X ) is not well ordered in R, since it contains

the sequence 1; 1=2; 1=3; ... .

In the pro of of next theorem wemakeuseofKunzi's characterization of the

spaces that admit a unique compatible quasi-unformity,[6].

3.2 Theorem. For a topological space X the fol lowing statements areequiva-

lent:

(i) X admits a unique compatible quasi-uniformity.

(ii) For every f 2 LS C (X );f(X ) is wel l ordered and compact.

(iii) For every f 2 LS C (X );f(X ) is wel l ordered and closed.

Proof. If X has a unique compatible quasi-uniformity,thenitisshown in [6]

that X is hereditarily compact. Let f 2 LS C (X ). By Prop osition 3.1, f (X )iswell

ordered. Also, since the quasi-uniformity SC is totally b ounded, f (X ) is b ounded,

by Prop osition 2.4.

We show that f (X ) is closed. Let (f (x )) be a strictly increasing sequence

convergentto t 2 R (if t were the limit of a non-increasing sequence in f (X ), we

would clearly have t 2 f (X )). For each n,letG = f x 2 X : f (x) >f(x ) g. Then

n n

Some prop erties of semi-continuous functions . . . 15

(G ) is a strictly decreasing sequence of op en sets whose intersection is f ([t; 1)).

If t=2 f (X ), wehave

G = f ((t; 1))

n=1

which is op en, thus contradicting Kunzi's result. This shows that (i) implies (ii).

The implication (ii) implies (iii) b eing obvious, weshow that (iii) implies (i).

From Prop osition 3.1, we have that X is hereditarily compact. Supp ose (G ) is

a strictly decreasing sequence of op en sets with op en intersection. For each n,

G , resp ectively. Let let g , g denote the characteristic functions of G ,

n n n

n=1

f = sup(1 1=n)g , h = max(f; 2g ). The functions g , g , f and h are all in

n n

LS C (X ), but h(X ) is not closed since it contains the sequence (1 1=n) and

1 2= h(X ).

4. The semi-continuous quasi-uniformity and Fletcher's conjecture

Fletcher [1] conjectured that a space with unique compatible quasi-uniformity

should have a nite top ology. Simple counterexamples to the conjecture can be

found in [3] and [8]. Nevertheless, it is shown in [2] that the assertion remains valid

for large classes such as the Hausdor or regular spaces.

In this section we see that the semi-continuous quasi-uniformity SC suces to

extend the validityofFletcher's conjecture to the classes of quasi-regular and R

spaces.

A top ological space X is said to b e quasi-regular whenever every non-empty

op en set contains the closure of some non-empty op en set. This class of spaces is

intro duced in [4] for the study of Baire like prop erties. The next theorem provides

a parallel result to the one stated in [2, Th.2.36].

4.1 Theorem. In a quasi-regular space X the fol lowing statements areequiv-

alent:

(i) X has a unique compatible quasi-uniformity.

(ii) The quasi-uniformity SC is total ly bounded.

(iii) Every semi-continuous function is bounded.

(iv) X is hereditarily compact.

(v) The topology of X is nite.

Proof. The implications (i) ! (ii) and (ii) ! (iii) are evident from previous

comments and results.

Weshow that (iii) implies (iv). Supp ose X is not hereditarily compact. Then

there is a strictly increasing sequence (G ) of op en sets. Making use of the quasi-

regularityof X , a sequence (H ) ofproperopensetsmay b e constructed inductively

such that

[

clH ; n 2: clH G ; and clH G

i 1 1 n n

i=1

16 J. Ferrer, V. Gregori, I. L. Reilly

For each n, let h be the characteristic function of H . Then, the function f =

n n

sup(n h )isanunb ounded elementof LS C (X ).

Since the implication (v) ! (i) holds in general [2, Th.2.36] weneedonlyshow

that (iv) implies (v). For each op en set G we assert the existence of a minimal

prop er clop en set contained in G. If not, we could de ne inductively a sequence

(H ) of prop er op en sets such that

H G; and clH H ; n 2

1 n n1

where the inclusions are strict. Thus the set

[

A = (H clH ) would not b e compact:

n n+1

Now supp ose the top ology is in nite. We construct inductively a sequence (H )of

minimal prop er clop en sets such that, for each n; H is disjoint with H [ H [

n+1 1 2

[ H .

We start out with any minimal prop er clop en set H , and assume H ; ...;H

1 1 n

are already de ned. Obviously, the sets H ;H ; ...;H do not cover X , otherwise

1 2 n

their minimalitywould turn them into a base for the top ology,contradicting our

assumption that the top ology is in nite. Thus, there is a minimal prop er clop en

set H contained in X (H [ H [[H ).

n+1 1 2 n

The existence of the sequence (H )contradicts the fact that X is hereditarily

compact since

[

H cannot b e compact.

n=1

It is shown in [2, Th.3.22] that, for Hausdor spaces, total b oundedness of SC

is equivalenttohaving a nite top ology. We pro ceed to extend this result to the

class of R spaces. Recall that a top ological space is R whenever clfxg 6=clfy g

1 1

implies the existence of disjoint neighbourhoods of x and y .

4.2 Lemma. Let X beanR spaceandsetA = f x 2 X :clfxg is open g. If

X A is a non-empty clopen set, then thereisa sequence (H ) of pairwise disjoint

proper open sets in X A and thus in X .

Proof. Let x 2 X A, then clfxg & X A. Thus, there is y 2 X A

such that clfxg 6=clfy g. Since X is R , there exist H ;V which are disjointopen

1 1 1

neighb ourho o ds of x; y , resp ectively,contained in X A. Since cl H isaproper

subset of X A, there is p 2 (X A) cl H ,andclfpg(X A) cl H . Thus,

1 1

there is q 2 (X A) cl H with clfpg 6=clfq g. As b efore, there are H ;V , disjoint

1 2 2

op en neighb ourho o ds of p; q ,contained in (X A) cl H , and H \ H = ,such

1 1 2

that (cl H [ cl H ) is a prop er subset of X A. Rep eating this pro cess inductively

1 2

one obtains the sequence (H ) of pairwise disjoint prop er op en sets.

4.3 Theorem. In an R space X , the quasi-uniformity SC is total ly bounded

if and only if the topology of X is nite.

Some prop erties of semi-continuous functions . . . 17

Proof. Suciency is certainly true in general. For the necessity part, consider

the set A as in the previous lemma. Then A 6= , otherwise the sequence (H )

of pairwise disjoint prop er op en sets would provide a function f = sup(n g )(g

n n

being the characteristic function of H ) which is lower semi-continuous and not

b ounded, contradicting Prop osition 2.4.

Furthermore, the collection f clfxg : x 2 A g must be nite. On the other

hand, we could cho ose an in nite sequence (clfx g) of pairwise disjointopensets,

thus yielding a contradiction as b efore. Therefore, since A = f clfxg : x 2 A g,it

is a clop en set and so is X A. By previous lemma, X A = . Thus A = X ,and

the nite collection f clfxg : x 2 A g is a base for the op en sets, thereby de ning a

nite top ology.

Notice that, under the assumptions of the last theorem, the nite top ology

obtained is strongly zero-dimensional.

4.4 Corollary. For regular (Hausdor ) spaces, SC is total ly bounded if and

only if the topology (the ground set) is nite.

It is shown in [8] that a top ological space X admits a unique compatible quasi-

uniformity if and only if every compatible quasi-uniformity is totally b ounded. We

havejust seen that, for R spaces, total b oundedness of SC suces. However, if

we remove the R assumption, it is no longer true that SC totally b ounded implies

a unique quasi-uniformity. As we see in the following example, it do es not even

imply the weaker prop ertyofhaving a unique quasi-proximity.

4.5 Example. Let ! denote the rst uncountable ordinal. In the ordinal

space X =[1;! ], consider the quasi-pseudometric:

d(x; y )=0 if x y; d(x; y )=1 otherwise;

0 0

which generates the bitop ological space (X; T ; T ), where T and T are the top olo-

gies induced by d and its conjugate d (x; y )=d(y; x).

It is easy to see that every T -semi-continuous function is b ounded, and the

0 0

same applies for T . Thus, their semi-continuous quasi-uniformities SC and SC are

b oth totally b ounded.

Nevertheless, from Theorem 1 of [5], (X; T ) admits more than one compatible

quasi-proximity, since the family of op en sets [1;x], x

is closed under nite intersections and unions. Also, (X; T ) admits more than

one quasi-uniformity, since the FINE quasi-uniformity formed by all sup ersets of

U = f (x; y ):x y g is not totally b ounded.

18 J. Ferrer, V. Gregori, I. L. Reilly

REFERENCES

[1] Fletcher, P., Finite topological spaces and quasi-uniform structures, Canad. Math. Bul. 12

(1969), 771{775

[2] Fletcher, P. and Lindgren, W.F., Quasi-uniform Spaces, Marcel Dekker, New York 1982

[3] Gregori, V. and Ferrer, J., Some classes of topological spaces with a unique quasi-uniformity,

Canad. Math. Bul. 29 (1986), 446{449

[4] Haworth, R.C.and McCoy, R.A., Bairespaces, Dissert. Math. Warszawa1977

[5] Kunzi, H.P., Topological spaces with a unique compatible quasi-proximity,Arch. Math. 43

(1984), 559{561

[6] Kunzi, H.P., Topological spaces with a unique compatible quasi-uniformity, Canad. Math.

Bul. 29 (1986), 40{43

[7] Kunzi, H.P., Some remarks on quasi-uniform spaces, Glasgow Math. J. 31 (1989), 309{320

[8] Lindgren, W.F., Topological spaces with a unique compatible quasi-uniformity, Canad. Math.

Bul. 14 (1971), 369{372

[9] Pervin, W.J., Quasi-uniformization of topological spaces, Math. Ann. 147 (1962), 316{317

[10] Stone, A.H., Hereditarily compact spaces, Amer. J. Math. 82 (1960), 900{916

(received 10.04.1995.)

Jesus Ferrer

Dpt. Analisis Matematico

Facultad De Matematicas

46100 BURJASOT (Valencia), SPAIN

Valentin Gregori

E.U. Informatica

Universidad Politecnica De Valencia

Aptdo. 22012

46071 VALENCIA, SPAIN

Ivan L. Reilly

Department of Mathematics and Statistics

UniversityofAuckland

Private Bag

AUCKLAND, NEW ZEALAND