0. Introduction and basic definitions and notation.

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria Г: PRACE MATEMATYCZNE XXVI (1986)

H elena P awlak and R yszard P awlak (tôdz)

On some properties o f closed functions in terms o f their levels

Abstract. The paper deals with closed real functions defined on a topological space. Some results are given concerning the restrictions o f such functions to supersets o f the union of their closed or connected levels.

0. Introduction and basic definitions and notation. The paper contains results concerning restrictions of real closed (or closed and connected) functions defined on some topological spaces. In particular, Theorem 2.3 is the next answer to the problem raised by Garg in [2] (partial answers are contained in [3], [4], [5]). This result is independent of theorems contained in mentioned papers.

We use the standard notions and notation, which were used in the paper by Garg [2]. In particular, if * is any property of sets, let £*(/) denote the union of those levels of f which have property *, and let Y*(/) be the set of elements ae Y for which the level / -1 (a) has property *. These sets S* (/) and Y*(/) may be called the domain and the range of levels of / with property *, respectively (see [2]), and it is clear that S*(/) = / -1 (**(/))•

The properties of being closed and connected are denoted in turn by к and c, respectively.

The subnet of the net {xe}aeS, where q> is a function mapping Z' into Z, such that for every a0e Z there exists o '0 e Z' such that:

if o' ^ a'0, then (р(а')Ь- o0

(where ^ is the directing relation in Z' and £- is the directing relation in Z), shall denoted by \ха>)а-еГ (see [1]).

By R we shall denote the set of real numbers with the natural topology.

The closure and the derived set of a set A we denote by Я and Ad, respectively.

The symbols (a, b), (a, b], [ a, b) and [a, b] for a < b or b < a denote (respectively) open, left-sided open, right-sided open and closed intervals. We denote by f ~ 1(a, b), b], f ~ x [a, b) and / _1 [a, b] the inverse images of those intervals to avoid superfluous double brackets.

6 — Prace matematyczne 26.1

The symbol A — lim xa will denote the limit of the net {x ^ }^ in the

oel

subspace A of X.

Finally we assume the following definition.

D efinition . We say that a function f : X -> Y, where X, Y are arbitrary topological spaces, is closed if f ( K ) is a closed set for every closed set К czX.

1. Closed levels and continuity.

D efinition 1.1. We say that a topological space X possesses property (H) if for an arbitrary set A a X and for every point xe/f there exists a net

!-vffUei P°ints ° f the set A such that xelim xa and (У {х,})* c (xj.

o e l oel

T heorem 1.2. Let X be a Hausdorff space possessing property (H) and let f : X -> [0, 1] be a closed function. Then flsjjj) is a continuous function.

Proof. Put S = Sk(f). It is sufficient to prove that for every net !х„У е1 of points of S, f (S — lim xa) = lim f ( x a).

oel oel _

_

Let {xa}ael be an arbitrary net of points of the set S and let x — S —

— lim x,,. Then also x = X — lim xff. We shall prove that

<теХ <x

f ( x ) = lim f { x a).

(tel

Now, assume to the contrary that /(x) is not a limit point of the net

\f{x„))ael. Thus there exists a subnet \xff>}a-er of {xe}eeS such that:

(1) (/(x) — r.,/(x) + fi) n [0, 1] n { f {xa’)\ o ’ e Z'} = 0 for some £ > 0.

Now consider two cases:

1° for every o' eZ\ x„ e S.

2’ there exists <т'еГ' such that xa.$S.

P r o o f o f case Г. Since (xff-}a.eI. is a subnet of so x = lim xa.

________ о'еГ

and consequently, xe (J {xff-}. According to Definition 1.1 there exists a net

о 'е Г

\Уо"\о”еГ' ° f points of the set (J {xff-j such that x = lim ye.. and Jx]

o 'eT o"eX"

=) ( (J which means that the set (x] u (J (ya is closed.

o " e I " а "е Г '

According to 1° and (1) we infer that:

l 00 for every a"eX", ya"e S and

(Г) (/ (x )-£ ,/ (x ) + fi)n [0, l ] n { f ( y a"): a " e X " } —0 for some £ > 0.

Since / is closed, we deduce that (J {/(>V')j u {/(x)J is a closed set.

In v ie w o f this a n d a c c o r d in g to ( Г ) w e o b ta in th at y {f{ye")} is a closed

set.

N o w w e sh all p r o v e that y { / O v )j is an in fin ite set. In fact, assum e

t o th e c o n tr a r y th at У { / ( > v ) } is a fin ite set. T h e n , a c c o rd in g to 1°°, fo r e v e ry o"eE'\ f ( y a» ) e Y k( f ) an d c o n s e q u e n tly , f ~ l ( У { / С Ы } ) is closed,

a" e l "

becau se it is fin ite u n io n o f c lo s e d sets. O n the o th e r hand, x e 7 " l ( U { / 0 V ' ) } ) . \ / _ 1 ( U { / O v O })• T h e o b ta in e d c o n tr a d ic tio n p ro ve s

a "e l" a "e l”

that у { f { y e")} is an in fin ite set.

a" e l "

S in ce У { f { ya")) is an in fin ite set an d У | / ( j V ') } <= [0 , 1], w e in fer that ( у { f ( y a»)}Y Ф 0 L e t a e ( У { f {ye"))Y- S in ce У { / С Ы } is a

a" e l " a" e l " a” e l "

clo s e d set so :

(2) ae У { / ( » ) } .

a” e l ”

L e t E* = {a" e E ” \ f (yff”) # x) an d let ^ b e the re la tio n o n the d irected set E " . O f co u rse, E* w ith re la tio n ^ is a lso a d ire c te d set.

O f cou rse, E* cr E" an d {ya*}a*e£* is a su bn et o f f ya"}0"ey , w h ic h m eans that x = lim ya*.

S in ce ( U у { > v , } ) d c : { x } so U {

} u { x } is a c lo sed

a* el* a” e l ” a* el*

set an d c o n s e q u e n tly :

(31 и { / 0 v ) } U < / < * )}

a*el*

is a c lo s e d set, to o . S in ce

(4 ) U { / ( » ) ! = ( U ! / ( > V ' ) ! ) \ M

a*el* a” e l "

w e in fer th a t a e ( У { f { ya*)}Y- H en ce, a c c o rd in g to (4), w e d ed u c e that

U i / 0 v ) } is n ot a c lo s e d set, w h ic h m ean s (a c c o r d in g to (2 ) and ( Г ) ) that

otÿéf(x) an d c o n s e q u e n tly , y { / ( » ) } и { / ( x ) } is n o t a clo s e d set. T h is

c o n tr a d ic ts (3). T h e o b ta in e d c o n tr a d ic tio n en ds th e p r o o f o f case 1°.

P r o o f o f c a s e 2°. F o r an a r b itr a r y elem en t x ^ o f {хв>}в>еГ such that

xa. $S, w e d e n o te b y net P°ints ^{° f} set $ such that

г (®i> <Tl

= bm yô .

T h e n

f { * a\)

= lim

(5)

( i

In fact, we assume to the contrary that / (x ^ ) is not a limit point of /(Уд1%елв^ Then there exists a subnet {yôai)}ô'eA'^ of {Уд such that

( f ( x a'1) - s i f ( x e'1) + e ) n [ 0, 1 I n j / O v 1*): ô' eA' ^} = 0 for some e > 0.

Now, it is easy to see that the proof of (5) will be similar to that in case 1° (it is sufficient to put xa> x in place x and — in place {x^jver)-

Let B (x) denote the local base in x, consisting of open sets and let A = {(a', U): a 'e l ' л Ue B( x ) л x ^ e V} .

Now we define the relation -3 in A in the following way:

(ff'i,

U2) o o \ < V 2

Ui

U2

(where is the relation in the directed set L'). Of course, A is the directed set with the relation -3.

Now we define the net {zô}ôeA in the following way:

For every Ô = (<r\ U) eA if х0< e S, we put z0 — xa> ;

if xa>$S let zb denote an arbitrary element of {у(з

- belonging to U and such that / (**)£ (/ (x )- ie ,/ (x ) + ie ) n [0, 1].

According to (1), (5) and the condition xa> — lim y(b ’’ such an element z&

0eAa'

exists.

Remark, that

(6) zôeS for every 5eA and x = lim z0.

ôeA

In fact, let U0 e В (x) and let a'0 be an element of I ' such that xff< e U 0.We put

= (°o> L^o). If -3 à — {a', U), then U c U0 and consequently zôe U c= U0.

According to (6) and the condition ( f ( x ) — j r, , f ( x) + j f , ) n n [0, 1] n {/ (zô): ô e A } = 0 for some v, > 0, we can reduce our consideration to case Г (it is sufficient to put [z0)0еЛ in place of {x*.}*-^)-

C orollary 1.3. Let X be Hausdorff space possessing property (H) and let f : X -* [0, 1 ~]bea closedfunction. Then f is a continuous function if and only if the set Sk( f ) is dense in X.

Z Monotonicify of closed functions.

D efinition 2.1. If E c R, a point x e R is said to be a bilateral limit point of E if it is a limit point of E from both sides, and we call E bilaterally closed if it contains all of its bilateral limit points.

D efinition 2.2. We say that a function /: X Y, where X, Y are arbitrary

topological spaces, is connected if / (C) is a connected set for an arbitrary

connected set C e l

Garg in [2] considers, among others the following problem: Under what hypotheses on X and / the set Yc( f ) is bilaterally closed. The answer to this question is affirmative if X is a er-coherent space and / is a connected function or X is a Hausdorff space and / is a connected and relatively proper function (see [2], Theorem 3.3). The next theorem (Theorem 2.3) gives a successive answer to the above question.

Le m m a

A [2]. Let f : X -> R be a connected function defined on a connected and locally connected space X. Then if ot, fie Yc( f ) and <x < ft, then the set Z ' 1 [a, /?] is closed and connected.

T heorem 2.3. Let X be a normal, connected and locally connected space and let J : X -+ R be a connected and closed function. Then Yc (/) is a bilaterally closed set.

Proof. Assume to the contrary that there exists a bilateral limit point y0 of Yc (/) such that y0<£Yc (/ ). Then there exists two sequences {yn }> {у» } of points of Yc( f ) such that {y “ } increases to y0 and { y * } decreases to y0 as n->oo.

Since, for n = 1, 2, ..., y~, y * e Yc (/), we infer that / “ 1 [y “ , y„+ ] is a closed

00

set for n = 1, 2, ... and consequently, / ~ 1 (y0) = f] f ~ 1 [y,f, y * ] is a closed set,

n= 1

too.

Since у0ф Yc( f) , we have f ~ 1(y0) = A u B, where A and В are closed disjoint and nonvoid sets in f ~ 1 (y0). It is obvious that A and В are closed in X.

There exist two open sets U and V such that A c U, В с V and U n V = 0.

Of course,

U) / - 1 (Уо) c U и К

According to Lemma A, the set / - 1 [y “ , y * ] is a connected set (for

« = 1 ,2,...) and, moreover, f ~ l [y ~ , y„+] n U Ф 0 Ф f ~ 1 [y ", y„+] n V.

Hence, for every n = 1 ,2 ,... there exists an element qne f ~ l [y„~, y„+] \(U u V).

Thus we have defined the sequence {q„}%L i- 00

Write К = У {q„}. Since, qn$ U u V (for every n = 1, 2, ...) so according

n= 1

to (1) we have K n f ~ 1(yo) = 0. We remark that f(q„) e [y ~, y„+ ] (for n = 1 ,2,...); then lim f { q n) = у0ф/ (X ) and this contradicts the fact that /

П — ► 00 is a closed function.

Garg in [2] raised the following problem (Problem 3.11, p. 27): Under what hypotheses on X and / a connected function /: X R is monotone or weakly monotone on the closure of the union of all connected levels. Partial solutions are contained in the articles [3], [4] and [5]. Now we shall formulate and prove a theorem giving a successive answer to the question of К. M. Garg. Our result is independent of theorems in papers [3], [4] and [5].

Le m m a

В [2]. Let f : X -> R be a connected function defined on a connected

and locally connected space X. Then flsfjj) is a continuous function.

De f i n i t i o n

2.4. The function J : X Y, where X, Y are two topological spaces, is said to be quasi-monotone if / ~ 1 (C) is connected for every connected set С c f ( X) .

Th e o r e m

2.5. Let X be a normal, connected and locally connected space and let f : X -> R be a connected and closed function. Then f l j yj ) Is quasi-monotone.

Proof. Write S — Sc( f ) and g = f\s. We first prove that g is a weakly monotone function (i.e., every level of g is a connected set).

Assume to the contrary that there exists a point xeS such that g~1 ( g(x)) is not a connected subset of S. We put a = q(x). Of course,

(1) * * Y c{f).

Then g ~ 1 (a) = A u B, where A, В are disjoint, nonvoid, closed subsets of g ~ 1 (a), which means (according to Lemma B) that A and В are closed in S and so in X.

Let U and V be open sets in X such that A cz U, В с V and U n V = 0.

According to (1) and Theorem 2.3, we infer that a is an unilateral accumulation point of Yc(f). Assume for instance that a is a left-hand accumulation point of Yc( f) .

Let (z„) i denote a sequence of points of the set S\(U и V) such that / (z„) / a. The scheme of the construction of the sequence {z„}£L i fulfilling the above conditions is the following. Let xAe A and xBeB. Hence, there exist nets {xe) e6l, P°ints S such that lim x* = xA and lim y6 = xB. Since g is a

fie л

continuous function (Lemma B) we infer that lim g( xa) = a = lim g{y0)- We put

<те! ôeA

P = [g(xa): (теХ}и {.g{yô): ô g A)

Yc{f).

Since X is a locally connected space there exist open and connected sets WA and WB such that xAeWA czU and xBeWB czV. Write Q = f { WA) c\f{WB) n n ( — oo, a). Hence Q is a non-degenerate interval. Let t Xi EPnQ and z1 e / ~ 1 (x1)\(U и V). Suppose we have defined ab ..., a„_ x and Zj, ..., z„_ x.

Write = i|a — a„_±| and let a „ e P n ( a - r „ _ 1, a) and z„e f ~ i {ct„)\{U и V).

Continuing this procedure we obtain the sequence [zn}™= l .

GO —

Write К = [j {z „ j. Hence, according to our assumption, / (К ) is a closed n~ 1

set. On the other hand, X n / _1(a) = 0 because К n f ~ 1 (a) c: S. Thus a this contradicts the closedness of f ( K ) .

The obtained contradiction ends the proof of the fact that g is weakly monotone.

According to Theorem 6.1.11 of [1] (p. 433) we obtain that g is quasi­

monotone.

References

ГП R- H n g e lk in g , Topoloqia oqolna, Warszawa 1975.

[2 ] К. M. G a r g . Properties o f connected functions in terms o f their levels, Fund. Math. 97 (1977), 17-36.

[3 ] Z. G r a n d e , Les ensembles de niveau et la monotonie d'une fonction, Fund. Math. (1979), 9 12.

[4 ] R. J. P a w 1 à к, On monotonicity o f connected functions defined on the locally connected continua, Zesz. Nauk. U .F . 34 (1980). 85 99.

[5 ] —, On the continuity and monotonicity o f restrictions o f connected functions, Fund.

Math. 114 (1981), 91-107.

INSTYTUT MATEMATYK.I UNIWERSYTETU LÔDZKIEGO LÔDZ, POLAND