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

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

### K. S

ieklucke### (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*

*2*

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

*2*

* 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.

Th e o r e m.*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*

*t*

* 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*

*0*

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

*0*

### 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*

*1*

* of the point x\ *

*and a neighbourhood U*

*2*

* of the point x*

*20*

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

### (3) *Fr UveYfl* *for v — *

*1*

### ,2 ,

### (4) *f ( U ’) <= r* *for v = 1, 2.*

**252** **К. S i e k l u c k i**

*Let TJQ = U*

*1*

* 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*

*0*

* — f ( x 0) and * *either x*

*0*

* — xl or x*

*0*

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

*0*

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

*0*

* с X — U 1. *

*This is evidently impossible, for TJ1 is open.*

*B y virtue of (4), W*

*0*

* = 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*

*0*

*: 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*

*0*

*—f { Fr U0) * *such that dim^Fr W*

*0*

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

*0*

*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*

*0*

*) = f { X ) —f ( U*

*0*

*) 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*

*0*

* 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*

*0*

* and t e X — U 0. *

*Indeed, if either a eFr TJ*

*0*

* 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*

*0*

*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 *

**w n e**### VF

0* (n — *

1 ### ,

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

1*(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 , (*

^{8}

### 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

**w n e***W0 and Q c P r F 0; thus since TT*

0* is open, Q * TC

0
*= 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 + *

1*, 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.