A generalization of a theorem of S. Mazurkiewicz concerning Peano functions

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X II (1969)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X II (1969)

K. S

(Warszawa)

A generalization of a theorem of S. Mazurkiewicz concerning Peano functions

1. Introduction. Mazurkiewicz proved in [3] that if I denotes the segment [0 ,1] and/: I 1

is a continuous mapping of I onto the square l 2, then there exist at least three distinct points tx, t

and tz such that f { t x)

— f ( t z) = f ( t 3). The result is the best possible, since there exists a Peano function of the segment onto the square which is of multiplicity 3 [1].

We say that a metric compact space X has the property of Mazurkiewicz, and simply write X e $Щ, if for every continuous mapping /: X Y, where dim Y > dim X, there exist at least three distinct points xx, x 2, x ze X such that f ( x x) = f ( x 2) = f ( x a). W e shall prove that the class ЭД1 has some inductive properties; namely, we shall prove the following theorem:

2.

Let us suppose that X is a compact metric n-dimension- al space, every n-dimensional subset of X has an interior point, and to every point of X there exists an arbitrarily small neighbourhood U with dim F r U 1 and F

U еШ. Then Х е Ш.

P r o o f . Let us suppose the contrary, i.e. let us suppose that there exists a continuous mapping f : X — ^>Y, where d i m Y > w + l, such that р г (у) < 2 for every yeY. Let d im ^ Y ^ n + 1, where y

eY. By the inductive definition of dimension, there exists a neighbourhood V of the point y

such that

(1) dim Pr W > n for any neighbourhood W с V of y0.

Let f^ iy o ) = {xl, 4 ) ; we (l° not exclude, however, the case x\ = x2 0.

By the assumption, we can choose a neighbourhood U

of the point x\

and a neighbourhood U

of the point x

such that (2) dim Fr XT < n --1 for v = 1, 2

(3) Fr UveYfl for v —

,2 ,

(4) f ( U ’) <= r for v = 1, 2.

252 К. S i e k l u c k i

Let TJQ = U

w U2 and let W0 = Int f { U 0). W e shall prove that

(5) y0e W0.

Indeed, otherwise there would exist a sequence yne Y —f ( U 0) (n = 1, 2, ...) such that yn -> y0. Let xnef-l {yn) for n = 1 , 2 , . . . B y compact­

ness of X , we can assume (choosing a subsequence, if necessary) that xn x0. Hence yn = f ( x n) ->/(a?0). I t follows that y

— f ( x 0) and either x

— xl or x

= x\. B y symmetry, we can assume that x

= xl, i.e. xn -> x l e U \ On the other hand, xnef~x{yn) <= X — U

с X — U 1.

This is evidently impossible, for TJ1 is open.

B y virtue of (4), W

= In t f { V l w TJ2) c /(Z71) w/(Z72) с V. Hence, by (5) and (1),

(6) dim Fr F 0 > w.

Let us note that /(Fr U0) = / ( F r ( L '1 w U2)) c / (F r TJ1 w Fr U2) c c /(Fr U 1) w / (F r TJ2). Thus, conditions (2) and (3) imply that

(7) dim/(Fr ?70) < n — 1.

Since Fr Wo is a compact space satisfying (6), the set [Fr TVYJ?i,

= {y eFr W

: dim^Fr Ж0 = n} is w-dimensional (see [2], p. 66). Hence, by (7), [F r TF0W ( F r U0) Ф 0, i.e. there exists a point p eF r W

—f { Fr U0) such that dim^Fr W

= n. Let Q be a closed neighbourhood of p in Fr W

which is disjoint with /(Fr UQ). Then, evidently

(8a) Q is compact,

(8b) Q c F r f 0- / ( F r U0),

(8c) dimQ > n.

I f qeQ, then qeFv W„ = Fr In t/ ( ?70) c Fr/( U0). ‘ Thus, there exist sequences sne f { U0), tne Y —f ( U

) = f { X ) —f ( U

) c f ( X — U0) for n = 1 , 2 , . . . such that q = limsn = Пт£л. There exist, therefore, sequences

П 71

(rne U0, t ne X — U

such that /(cr„) = sn, f ( r n) = t n for n 1, = 2 , . . . By compactness of X we can assume that an a and t n -> r, hence

*»-*■/(<*) and and we infer that f ( a) = q = f ( r ) . Obviously ere t70 and t e X — U0. In fact, however, we have a e U

and t e X — U 0.

Indeed, if either a eFr TJ

or teFr U0, then q = f ( a) — f { r ) c /(Fr Z70), contrary to (8b). In particular, we infer that а Ф r. Thus we have shown that any point of Q is the image under / of two distinct points: one from TJ

and the other from X — U0. I t follows that / (Q) = A w B, where 0 ф A

<= U 0, 0 Ф В a X — U0. B y (8a), / (Q) is compact, and since / X{Q)r\

Fr TJQ = 0, we infer that A and В are compact. Since f \A and f \B

are one-to-one, they are homeomorphisms onto Q. In particular, we infer

from (8c) that

Generalization of a theorem of Mazmhiewicz 253

W e shall prove that A is a boundary set. Indeed, let aeA. Then f(a)eQ a Fr W 0 and there exists a sequence

VF

(n —

,

, ...) such that wn -»/(«-). Since W0 c f { U 0), the set /_

(геи) ^ U0 is not empty for every n. Let us choose un€f~1(wn) гл U0 for n = 1 ,2 , ... By compact­

ness of X we can assume that un -> щ. Then wn = f ( u n) ->/(w0), so that f ( u 0) — f(a ). Moreover, u0eU0, and since f ( u 0) — f(a )e Q , (

b) implies that in fact u0e U0. Since f ( u Q) = f{a ) and f\A, f\B are homeomorphisms, we infer that u0eA and щ = a. Thus un -> a. I t remains to note that un i А. И? however, uneA for some n, then wn = f ( u n) e f ( A ) — Q, which is impossible, for

W0 and Q c P r F 0; thus since TT

is open, Q TC

= 0. W e have therefore, proved that a is a boundary point of A.

Since A is a boundary set in X , by the assumption, dimJ. < w — 1.

But this contradicts (9) and the contradiction completes the proof of our theorem.

3. Corollaries. I f X is an w-dimensional polyhedron, then any u-di- mensional subset of X has interior points. Moreover, any point of X has an arbitrarily small neighbourhood whose boundary is an (n — ^-dimen­

sional polyhedron. Thus by a simple induction we get

Co r o l l a r y 1

. Any polyhedron has the property of Mazurkiewicz.

By the similar arguments we get

Co r o l l a r y

2. Any compact manifold has the property of Mazurkie­

wicz.

In other words, we have found the following generalization of the theorem of Mazurkiewicz: I f X is an n-dimensional polyhedron (resp. manifold) and f : X —^ Y , where dim Y ^ n +

, is a continuous mapping, then there exist at least three distinct points хг , х 2, х 3е Х such that f ( x x) = f ( x 2) = f { x 3).

The assumption that X is a polyhedron (resp. a manifold) is in a sense essential as it is shown by the example of a Cantor set C and the well- known function of C onto I.

R e fe r e n c e s

[1] H. H a lm , Ann. Mat. Ser. I l l , 21 (1913), p. 33.

[2] K. K u r a t o w s k i, Topologie I I , Warszawa 1952.

[3] S. M a z u r k ie w ic z , Prace Mat. Piz. 26 (1915), pp. 113-120.