ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXIV (1984)
Stanislaw T. Czuba (Wroclaw)
Some other characterizations of point wise smooth dendroids
In this paper we give three new characterizations of the pointwise smooth dendroids. The first by using components of some open sets, the second — with the set function К defined by Jones (see [8], p. 404), and the third using the concept of semi-aposyndesis of continua.
The author is very much indebted to Professor J. J. Charatonik for his advice during the preparation of this paper.
A continuum means a compact connected metric space.
A continuum X is said to be
— arcwise connected if for every two points a and b of X there exists an arc ab joining a with b and contained in X .
— hereditarily unicoherent if the intersection of any two subcontinua of X is connected (or empty),
— a dendroid if it is arcwise connected and hereditarily unicoherent.
The closure of a set A <= X is denoted by Â, and we put Fr A = Â n X \A for the boundary of A and Int A — X \X \A for the interior of A .
Given a sequence of subsets A n of a topological space X , we denote by LiA„ (LsA„) the set of all points x of X for which every neighbourhood intersects A n for almost all n (for infinitely many n). If Li A„ = Ls A n, then we say that (he sequence A n is convergent and we put LtA„ = LiA„ = LsA„.
If a continuum К is contained in a set A, then by C(A, К ) we denote the component of the set A which contains K.
Given a set A contained in a continuum X , we define T(A) as the set of all points x of X such that every subcontinuum of X which contains x in its interior intersects A. We write simply T(a) instead of T({a}).
De fin it io n I (see [5], Definition 2, p. 216). A dendroid X is said to be pointwise smooth if for each point x of X there exists a point p(x) of X such that for each sequence x„ convergent to a point x we have Lt x„p(x) = xp(x). A point p(x) will be called the initial point for x in X .
Theorem 2. Let X be a dendroid. The following three conditions are equivalent :
(a) X is pointwise smooth,
(b) for each point x o f X there exists a point y o f X such that for each continuum К а X which contains an arc xy and for each open set U a X such that K a U we have x e l n t C(U, K),
(c) for each point x o f X there exists a point у o f X such that for each open set U а X such that xy c= U we have x e ln tC ( U ,x ) .
P ro o f. The implication (a) =>(b). Let x be a point of the pointwise smooth dendroid X and let К a X be continuum containing the arc xp(x). Let U denote an open set which contains K . If x ^ In t C(U , K), then there exists a sequence of points x ne X \C { U , K) and converging to x. Observe that x„p(x)\U Ф 0 . Thus Lsx„p(x) n Fr U Ф 0 . A simple calculation shows us that Fr U n K = 0 . From the pointwise smoothness of X we have that Lsx„p(x) = xp(x) c= K . This is a contradiction.
The implication (b)=>(c) is trivial.
The implication (c) =>(a). Let x be a point of X and let x„ be a sequence convergent to a point x. If Ls x ny Ф xy, then let k e Ls x ny \xy and let U а X be an open set such that k e Ü and xy a U . It is easy to see that x n$C {U , x) so x ^ In t C(U, x), a contradiction. Since Lix„y is a continuum containing x and у (see e.g. [1], Lemma 1, p. 6), we have Lix„y = Lsx„y = Ltx„y = xy.
So we can take the point у as p(x) — the initial point for x in X . The proof is complete.
It is proved in [6] that
Pr o po sit io n 3 (see [6], Theorem 3.1, Corollary 3.2, p. 198 and 199). The following properties are equivalent for a dendroid X :
(a) X is pointwise smooth,
(b) for each pair o f different points x and у from X we have T(x) n x y = {x}
or T ( y ) n x y = (y}, or T ( x ) n T(y) = 0 ,
(c) for each point x o f X there exists a point p(x) o f X such that (i) for each sequence x„ convergent to a point x we have Ltx„p(x) = xp(x) and
(ii) x eT(p(x)).
As an easy consequence of the definition of pointwise smoothness of dendroid we have the following
Pr o po sitio n4. A dendroid X is pointwise smooth if and only iffor each point x o f X and for each sequence x n o f points on X converging to x there is a point q o f X such that Lsx„x = xq and for any initial point p(x) for x in X we have xq a xp(x).
F. Burton Jones in his paper [8] defines the following set function.
De fin it io n 5 (see [8], p. 404, cf. [9], p. 373). Given a set A contained in a
continuum X we define К (A) as the set of all points x of X such that there do not exist an open set U and a continuum H such that A cr U a H cr X Jx}.
It is proved in [9] (Lemma 1, p. 374) that if X is hereditarily unicoherent and if A is connected, then К {A) is a continuum. We write simply К (a) instead of K({a}).
Lemma 6. For any subcontinuum A o f the continuum X we have К (A)
= {x: T { x ) n A Ф 0}.
P ro o f. Let a point x of X be such that T ( x ) n A ф 0 . It means that there exists a point у of A such that y e T(x). Therefore for each continuum C which contains у in its interior we have x e C . Thus for each continuum D such that A c i l n t D we have x e D . So, x gK (A).
Now, let x be such that T ( x ) n A = 0 . It means that for each point a of A there exists a continuum Da such that a e ln tD a and x £ D a. The family {I n tDa:
a s A} covers the compact set A, so we can take finitely many sets Da i, . . . , D„n
П П
to cover the set A. It is evident that A c Int( (J Da.) and x ^ (J Da.. Thus
i=l ' i=l ‘
хф К (А ). The proof is complete.
Let us note that, using a set function T* in the sense introduced in [7], p.
103, one can reformulate Lemma 6 as: К (A) = T*(A) for any subcontinuum А с X .
Co r o l l a r y 7. For any two points x and у o f a continuum К we have x e K ( y ) if and only if y e T ( x ) .
Using the set function К we have the following characterization of pointwise smooth dendroids.
Th e o r e m 8. A dendroid X is pointwise smooth if and only if for each point x from X the set K (x) is an arc xy, where у can be taken as an initial point for x in X
(y = p{x) in the sense o f Proposition 3 (c)).
P ro o f. Let a dendroid X be pointwise smooth and let p(x) denote the initial point for x in X (from Proposition 3 (c)). It follows from the definition of K (x) and from Corollary 7 that xp(x) c= K(x). If К (x) \xp(x) Ф 0 for a certain x from X , then let у eK (x)\xp(x). Consider the following three cases.
C a s e 1. If xeyp(x), then T(y) n y p (x ) з xy ф [y], T(p(x)) n yp(x) з xp(x) Ф |p(x)} and х еТ (у) n T(p(x)) Ф 0 (see Corollary 7), so by Proposition 3 the dendroid X is not pointwise smooth.
C a s e 2. If p (x )e x y , then it follows immediately from Proposition 4 that p(x) is not an initial point for x in X .
C a s e 3. If y x u x p (x ) is a triod, then let y x u x p (x ) = xt u p (x )f u y f, where t is the centre of this triod. Therefore T(y) n yp{x) zj yt Ф (y}, T (p(x))r\yp(x) 3 p(x)t Ф |p(x)| and T(y) n T (p (x )) з \t] ф 0 (see Corol
lary 7), so the dendroid X is not pointwise smooth. Thus K(x) = xp(x).
Now, let there exist two points a and b of X such that T (a )n a b Ф {a}, T(b) n a b Ф {b} and T(a) n T(b) Ф 0 ; then from hereditary unicoherence of X
we have a point гф {а,Ь} such that z e T ( a ) n T(b) n ab. Let us observe that a and b both belong to K (z), so the set К (z) is not an arc with z as one of its end points. This is a contradiction, which finishes the proof.
As an easy consequence of Proposition 3 (c) and Theorem 8 we have the following
Corollary 9. I f X is a pointwise smooth dendroid, then for each point x o f X there is exactly one point p(x) (namely the end point o f the arc К (x), distinct from x) such that
(i) for each sequence x„ convergent to a point x we have Ltx„p(x) = xp(x) and
(ii) x e T (p (x )).
This point will be denoted by Pi(x).
Corollary 10. I f X is a pointwise smooth dendroid, then no point from the open arc xpx (x) = K (x )\{x , pl (x)} is the initial point for x in X .
The following theorem show us relations between functions T and К on pointwise smooth dendroids.
Theorem 11. I f a dendroid X is pointwise smooth, then for each pair o f points x and у o f X we have
(i) K (x) = К (у) if and only if x = y, and (ii) T(x) = T(y) if and only if x = y, and (iü) T(x) n К (x) = {x ], and
(iv) ^ ( T ( x ) ) = T ( x ) u K ( x ) , and (v) K (T (x)) <= T(K(x)).
P ro o f, (i) If K(x) = К (у), then by Theorem 8 it follows К (x) = xp(x) and K(y) — yp(y), so x = у or x = p(y). If x = p(y), then K(x) = K(y) = xy, thus T (x )n x y = T(y) n x y = xy and T ( x ) n T(y) zd xy (see Corollary 7), so by Proposition 3 we have x = y.
(ii) Let T(x) = T(y); then T ( x ) n T(y) =э xy and T (x )n x y = T (y )n x y
= xy, so by Proposition 3 we have that x = y.
(iii) If y e T(x) n K (x) and у Ф x, then y e T ( x ) and x e T ( y ) (see Corollary 7), so T(x) n T ( y ) xy Ф 0 , T(x) n x y = T(y) n xy = xy. This contradicts Proposition 3.
(iv) It follows from definitions that T(x) u K ( x ) <= K(T(x)). Let y e K( T ( x ) ) . It means that T(y) n T(x) Ф 0 (see Lemma 6). Using Proposition 3 we have that T(x) n x y = {x} or T(y) n x y = {y}. If T(x) n x y = (x), then by hereditary unicoherence of X we have that x e T ( y ) , whence y e K ( x ) (see Corollary 7). In case T ( y ) n x y = (y} the proof is similar.
(v) This inclusion follows immediately from (iv).
The proof is complete.
Some simple examples show us that the assumption in Theorem 11 that the dendroid X is pointwise smooth is essential.
If we take a one-point union of two harmonic fans (see [2], Proposition 3, p. 110) we obtain a dendroid which has all properties (iH v) from the theorem above but which is not pointwise smooth.
We say that the continuum X is aposyndetic at x with respect to y if there exists a continuum H such that x e l n t H a H c X\{y}. Recall that the continuum X is said to be semi-aposyndetic if for each pair of distinct points, X is aposyndetic at one of them with respect to the other.
It is easy to prove that we have the following
Pro po sitio n 12. A hereditarily unicoherent continuum (thus a dendroid) X is semi-aposyndetic if and only iffor each pair o f distinct points x and y o f X we have either уфТ( х) or хфТ(у).
Defin it io n 13. A non-empty subcontinuum К Ф X of a dendroid X is called an R 3-continuum in X if there exist an open set U such that K cr U, and a sequence C„ of components of U such that Li Cn = K .
It is proved in [6] that
Propo sitio n 14 (see [6], Proposition 3.7, p. 201). I f a dendroid X is pointwise smooth, then it contains no R 3-continuum.
It is proved in [3] that
Pr o po sitio n 15 (see [3], Corollary 8, p. 306, cf. [4], Corollary 11, p. 78). I f X is a dendroid and A and В are closed subsets o f X such that A n T ( B ) = 0
= T(A) n B and T ( A ) n T ( B ) Ф 0 , then the dendroid X contains an R 3- continuum.
Now, we have the following characterizations of pointwise smooth dendroids.
Theorem 16. Let X be a dendroid. The following conditions are equivalent : (a) X is pointwise smooth,
(b) X is semi-aposyndetic and X does not contain an R 3-continuum, (c) X is semi-aposyndetic and X does not contain two points x and у such that хфТ(у), уфТ( х) and Т ( х ) п Т ( у ) Ф 0 .
P ro o f. Let X be a pointwise smooth dendroid. It follows from Proposition 3(b) and Proposition 12 that X is semi-aposyndetic, and by Proposition 14 the implication (a) => (b) holds.
The implication (b)=>(c) follows from Proposition 15.
To prove the implication (c) =>(a) let points x and у from X be such that T (x)nxy ф Jxj and Т ( у ) п х у Ф [у}. If T (x)n T(y) Ф 0 , then by hereditary unicoherence of X we have T (x) n T (у) n xy ф 0 . If T(x) n T { y ) n x y = {p}, then it is easy to see that рф {x, y}, so хфТ(у) and уф T(x), a contradiction. If T(x) n T (y )n x y = pq, where p Ф q and pe x q , then from q e T ( x ) it follows that qeT( p) , and from p e T ( y ) it follows that peT(q), so X is not semi- aposyndetic by Proposition 12. Hence T (x )n T(y) = 0 and by Proposition 3(b) we have that X is pointwise smooth. The proof is complete.
References
[1] J. J. C h a r a to n ik , On fans, Dissertationes Math. (Rozprawy Mat.) 54, Warszawa 1964, 1-40.
[2] —, Contractibility and continuous selections, Fund. Math. 108 (1980), 109-118.
[3] S. T. C zu b a, The set function T a n d R-continuum, Bull. Acad. Polon. Sci. Ser. Sci. Math. 27 (1979), 303-308.
[4] —, R l-continua and contractibility, Proc. Inter. Conf. Geom. Topol. Warszawa 1978, PWN, 1980, 75-79.
[5] —, A concept o f pointwise smooth dendroids, Uspehi Mat. Nauk. 34 (6) (1979), 215-217; or Russian Math. Surveys 34: 6 (1979), 169-171.
[6] —, On pointwise smooth dendroids, Fund. Math. 114 (1981), 197-207.
[7] H. S. D a v is and P. M. S w in g le , Extended topologies and iteration and recursion o f set- functions, Portugal. Math. 23 (1964), 103-129.
[8] F. B. J o n e s, Concerning non-aposyndetic continua, Amer. J. Math. 70 (1948), 403-413.
[9] E. J. V o u g h t, Monotone decompositions o f H ausdorff continua, Proc. Amer. Math. Soc. 56 (1976), 371-376.
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