On preordered topological spaces and increasing semicontinuous functions

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X I (1968)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1968)

Z. Semadeni and П. Zidenberg (Poznań)

On preordered topological spaces

and increasing semicontinuous functions

Let X be a preordered topological space (by a preorder we mean a relation < such that (i) the conditions x < у and у < z imply x < z, (ii) x < x for all x\ at the moment we do not assume any relations between the topology and order). A real-valued function / on X is called increasing

[decreasing] if x < у implies f(x) < № № ) ^ /( i/) ] | a subset A of X

is called increasing [decreasing] if its characteristic function %A is in­ creasing [decreasing]. The preorder is called discrete if x < у is equivalent to x = y) if this is the case, then every real-valued function on X is both increasing and decreasing.

X is said to be monotonically normal p) if it satisfies the following

condition: Given a closed decreasing set F and a closed increasing set H such that F rs II = 0, there exist an open decreasing set U and an open increasing set V such that

F c U, Я c 7 , U r ^ V = 0 .

It is easy to show (see [8], p. 29) that X is monotonically normal if and only if the following condition is satisfied:

if F is a closed decreasing subset of X and F is contained in an open decreasing set W, then there exists an open decreasing set U such that F c U and D(TJ) c W , where D(U) denotes the smallest closed decreasing set containing U.

If the preorder is discrete, then any subset of X is both increasing and decreasing and the above condition is just the ordinary definition of a normal topological space.

Proposition. Let X be. a preordered metric space with the distance function q satisfying the following conditions:

314 Z. S e m a d e n i and H. Z i d e n b e r g

(i) if A is a closed increasing subset of X and x < y, then q(x, A) > e (y ,A ),

СИ) if

в

is a closed decreasing subset of X and x < y, then q(x, B) < Q (y ,B ).

Then X is monotonically normal.

Proof. Let F , H be closed disjoint subsets of X , let F be decreasing and H increasing. Then the function

f( ) ^ F) = 1

q(x, F)-^ q(xу В) В) e(oo,F)

is continuous and increasing on X ; moreover, 0 < / ( # ) < 1 for a? in X ,

f(oc) = o for x in F, and f{x) — 1 for x in B . It is clear that the sets U = {x: f(x) < |} and V = {x: f(x) > |} are open, U is decreasing, V is increasing, F a U, В с: V, and JJ r\ V = 0.

Let B n denote the Euclidean n-space with the cardinal order defined by the condition: (xx, .. . , xn) ^ (Vi, iff x$ for % — 1, .. . , n. It is easy to show that B n satisfies the assumptions of the above propo­ sition; hence it is monotonically normal.

Theorem (2). The following conditions are equivalent:

(a) I f g and h are increasing real-valued functions on X , g is upper

semicontinuous and h is lower semicontinuous, and if g(x) < h(x) for x in X , then there exists a continuous increasing function f on X such that g(x) < / ( # ) < h(x) for x in X.

(b) X is monotonically normal.

If the preorder is discrete, then the above theorem is just the purely topological “between theorem”, which was proved by H. Hahn in 1917 for metric spaces (see [3], [4], [5]) and generalized in 1951 to the case of normal spaces independently by M. Katetov [6] and С. H. Dowker [2]. For increasing functions this theorem has been proved by ISachbin ([8], p. 116) under the additional assumption that X is compact. As a matter of fact, Nachbin assumed only that at least one of the functions g, h is increasing whereas we assume that both g and h are increasing. The following example shows that if X is not compact, this stronger assumption is not superfluous.

Let. X — B 2, let h be the characteristic function of the open half- plane {(x, y): x > 0}; then h is increasing and lower semicontinuous. Let g be equal to 0 in the left and upper closed half-planes; thus, if x < 0

Preordered topological spaces 315

or у > 0, then g( x, y) = 0; furthermore let g( x, y) = 1 when x > 0 and у < — 1/x, and let g be linear on each segment (0, y ) , ( —1 / x , y ) , where у < 0. Then g is continuous and g { x , y ) < h { x , y ) for (x , y) in R 2. I f / is any increasing function such that g(x, у ) У) < h( x , y) for (x, y) in R 2, then f ( x , y) = 0 if x < 0 and f ( x , у) — 1 if x > 0; hence

such an / cannot be continuous.

P r o o f of the theorem. Let us firs t assume (a). Let F, H be closed d isjoint subsets of X and let F be decreasing and H increasing. Then the set О = X \ F is open and increasing; moreover, H a G. Therefore

Xh is increasing and upper semicontinuous, %G is increasing and lower semicontinuous, and %h < %g- Th u s, by (a), there exists a continuous

increasing function / such that xh < / < Xg- I t is clear that f ( x ) = 1

for x in H and f ( x ) = 0 for x in F. Consequently, the sets U = {x: f ( x ) < and V — {x: f ( x ) > |} have the desired properties.

Let us now assume (b). Since there is an order-preserving homeo- morphism from the extended real line onto [0,1], we may assume that 0 < g(x) ^ Ть (*r) ^ 1 for x in L . The proof is sim ila r to that of Urysohn^s lemma. Let Q denote the set of rationals of [0,1]. I f qeQ, let

F{q) = { x e X : h{x) < q] and V(q) = {x e X : g(x) < q}.

I t is obvious that F( q) is closed, V(q) is open, and both are decreasing; moreover, if q < r, then

(1) F( q) c 7 ( r ) , F( q) c F( r ) , V(q) a V( r).

We shall show that for each q in Q we can find an open decreasing set

W(q) such that

(2) F( q) c= W( r) if q < r

(3) D( W( q) ) c: W( r ) if q < r

(I) D( W( qj ) a V{r) if q < r

Let qx, q2, ... be a sequence of a ll elements of Q such that qx = 0, q2 — 1,

qn ф qm if % Ф m. We proceed by induction. F irs t we define W (0) = 0 ,

TF(1) = F ( l) , and then we suppose that the open decreasing sets

W{ q x) , . . . , W( q k) have been defined and sa tisfy (2), (3), (4) for q, r in Tk = {#i > • • • ,Ук] • I f we denote

p = sup {q: q e T k & q < tk+1} and p = in i{q : q e T k & q > t k+1},

then, by (1), F ( p ) c= V(p' ). Moreover,

F ( p ) c= W( p ' ) , D ( W( p ) ) cz W ( p ' ) , and D ( W{ p ) ) c V(p' ).

3 1 6 Z . S e m a d e n i a n d H . Z i d e n b e r g

decreasing set

U

such that

F(p)

w

D(W(p)) a U

and

D(U)

<=.

W(p')

V (p').

T h is set

U

w ill be denoted by

W (qk+

1). I t is obvious that conditions (2), (3), (4) are satisfied for

q

in

Tk+1.

Let ns now denote f ( x ) = sup{g: qeQ & x 4 W(#)} for x in

X.

I t is clear that 0 < /(# ) < 1. We shall prove the continuity of / at an arbitrary point x of

X.

Le t ns f irs t consider the case where 0 < f { x ) < 1. Let r, s be any rationale such that 0 < r < f ( x ) < s < 1. Then x e W { s ) . I f r < q < f { x ) , then x4W{q) - , consequently, x 4D{W{r ) ) . There­

fore U = W ( s ) \ D ( W ( r j ) is a neighborhood of x. Let y e U . Then ye W($); hence s < f ( y) . S im ila rly , y 4 D( W{ r ) ) implies у 4 W{r) and f ( y ) > r.

We have shown that y e U implies r < / (y ) < s. T h u s , / is continuous at x. I f /(a?) = 0 or f ( x ) = 1, the proof is sim ila r.

The function / is increasing. Indeed, let x ^ . y and q e Q . Since

X\W(q)

is increasing, the set

{q: x4W(q)}

is contained

{q: yiW{q)}.

Consequently,

f(x)

_^

f(y)-A ll that remains to be shown is that

g(x)

</(ж) ^

Ji(x)

for

x

in

X.

Suppose, if possible, that

f(x)

>

h(x).

Then there exist

q, r

in Q such that

f(x)>r>q>h(x).

Consequently,

xiW{r)

and

xeF{q).

Th is contradicts (2). The proof of the inequality

g(x)

</(a?) is sim ila r. T h is concludes the proof of the theorem.

Le t us note that the “between theorem” is also true if

X

is downward filte rin g (i.e., such that for any

x,

у i n

X

there is a 0 in

X

such that г <

x

and z < y),

h

is lower semicontinuous and increasing,

g

is upper semi- continuous and decreasing, and

g

< then there exists a constant function / such that

g

< / <

h.

R eferences

[1] F. F. B o n s a ll, JSemi-algebras of continuous functions, Proc. Internat. Symp. on Linear Spaces, Jerusalem 1961, pp. 101-114.

[2] С. H. D o w k e r , On countably paracompact spaces, Canadian J. Math. 3 (1951), pp. 219-224.

[3] H. H ah n , Tiber halbstetige und unstetige Funktionen, Wiener Akad. Ber 126 (1917), pp. 91-110.

[4] — Theorie der reellen Funktionen, Berlin 1921.

[5] F. H a u s d o r ff, Tiber halbstetige Funktionen und deren Verallgemeinerung, Math. Zeitschr. 5 (1919), pp. 292-309.

[6] M. K a t e t o v , On real-valued functions in topological spaces, Fund. Math. 38 (1951), pp. 85-91.

[7] L. N a c h b in , Sur les espaces topologiques ordonnes, С. B, Aead. Sci. Paris, 226 (1948), pp. 381-382.

[8] — Topology and order, Princeton 1965.