Homotopically fixed arcs and the contractibility of dendroids
by
J. J. Charatonik and Z. Grabowski (Wroclaw)
Abstract. The concept of an R-are in a dendroid is introduced and studied in the paper, to
s.how a new class of non-contractible dendroids, namely of dendroids which contain an R-are.
Using this concept, hereditarily contractible fans are characterized as smooth ones, but the problem
of finding a characterization of hereditarily contractible dendroids remains open.
Although the theory of contractible spaces is well-known (see e.g. [4] and [6]),
general methods are not very useful for investigation of the contractibility of some
special spaces, e. g. of some curves. In particular there are only a few papers which
concern the contractibility of dendroids; no characterization of the class of contractible dendroids is known at present, and merely several conditions which are either
necessary or sufficient have appeared in the literature. This paper is a contribution
to the attempt to find further conditions concerning the contractibili'ty of dendroids.
All spaces considered in this paper are assumed to be metric. A continuum
means a compact connected space. A property of a continuum X is called to be
hereditary if each subcontinuum of X has tlrls property. A continuum X is said to
be arcH'ise connected if for every two points a and b of X there exists an arc ab
joining a with b and contained in X. A continuum X is called unicoherent if for each
two subcontinua A and B of X such that X = A u B the intersection A n B is connected. A dendroid means an arcwise connected and hereditarily unicoherent continuum. A point p of an arcwise connected space X is called a ramification point
of X provided there are three arcs pa, pb, pc such that p is the only common point
of every two of them. A dendroid which has only on~ ramification point is called
a fan. A point x of an arcwise connected space X is called an end pofnt of X if
it is an end point of every arc containing it and contained in X.
Given a space X with a metric d, let A be a subset of X and let e be a positive
number. We denote by Q(A, a) the spherical a-neighbourhood of A, i.e., the set
of all points x of X for which there exist points a of A with d(x, a)< e. We denote
by [a, fi] the closed segment of reals from a to fi, i.e. [a, /3] = {t: rx~t~[J}, and
we put I for [0, 1]. The closure of a set A eX is denoted by A, and we put
Fr A = A n X'-A for the boundary of A. To describe some examples we use the
symbol xy for the straight line segment joining the points x and y in the Euclidean
space. We hope there will be no confusion with the closure of a set.. Given a se4*
230
Homotopically fixed arcs and the contractibility of dendroids
J. J. Charatonik and Z. Grabowski
quence of subsets An of X, we denote by LimA, the topological limit of An in sense
n-~ooo
of [6], § 29, VI, p. 339.
.
.
A mapping means a continuous transformation. A mappmg H: Xx I-+ Y IS
called a homotopy. If X c Y and if H(x, 0) = x for each x eX, then the homotopy His said to be a deformation of X in Y. Furthermore, if for each x eX the
point H(x, I) is the same (i.e., if H( ·, 1) is a consta.nt mapping), then t~e deformation H is called a contraction of X in Y. They are simply called defonnat10ns and
contractions of X if y = X. If a contraction of X (in Y) exists, then X is said to be·
contractible (in Y) (see [4], I, 8, p. 11 and 12; cf. [7], §54, I, p. 360; IV, V and VI,
p. 368-375).
PROPOSITION 1. Let a space X contairi two sets A and B such that
(1)
A =1=0,
B =I=X
and
(2) for every deformation H: X xI-+ X we have H(A
x I) c
B.
Then X is not contractible.
In fact, suppose there exists a contraction H with H(A xI) c B which con·
tracts X to a pointy, i.e., H(x, 1) = y for every x eX. Thus X is arcwise connected
([7], §54, VI, Theorem 1, p. 374) and hence we may take an arbitrary point of X
as y. Taking y in X'.B and x in A we get a contradiction.
DEFINITION 2. A non-empty subset A of a space X is said to be homo topically
fixed if for every deformation H: Xxi-+X we have H(Axi)cA.
As an immediate consequence of the above definition and of Proposition 1 we get
PROPOSITION 3.lf a space X contains a proper subset A which is homotopically
fixed, then X is not contractible.
Let X be a dendroid. Consider an arbitrary arc ab contained in X and a point x
of X. If, for some number e>O we have x e Q(ab, e) and the intersection
xa n Q(ab, e) is non-connected, then we denote by x(e) the first point of the arc xa
ordered from x to a which lies in FrQ(ab,e), i.e., x(e)exanFrQ(ab,e) and
xx(e) n Fr Q(ab, e) = {x(e)}.
DEFINITION 4. An arc ab contained in a dendroid X is called an R~arc if
1o there are two sequences {u11 } and {v11 } of end points of X such that
lim ut, =a
and
lim
V11
= b;
The case of a degenerate RNarc (j. e. when a = b) is also acceptable.
THEOREM 5. Every R-are contained in a dendroid X is homotopically fixed.
Proof. Let a dendroid X contain an RNarc ab. Suppose, on the contrary, that
there exists a deformation H: Xxl-+Xfor which H(ab x 1)'-.ab :1= 0, i.e., for which
there exists a number t' e I with
·
H(ab x {t'})'\ab :1= 0.
(3)
Let a number e>O satisfy condition 2° of Definition 4. For. each point p e ab
put
tP = sup{t e I: H({p} x [0, t])c Q(ab, fe)},
and let t 0 = inf{tP: p E ab}. Observe that t0 >0. In fact, suppose t~ = 0. Then there
is a sequence of points Pn e ab such that the sequence of corresponding numbers tPn
tends to zero. It follows from the compactness of the arc ab that the sequence {p11}
contains a subsequence {p111J which converges to a point p 0 e ab. We see by the
definition of tP that if tp<l, then H{p, tp) e Fr Q(ab, fe). Since the sequence {tPnJ
tends to zero as a subsequence of {tP,J, we may assume tPnk < 1 for sufficiently large k,
and therefore H(pnk' tPn,) e Fr Q(ab, ts). The mapping H being continuous and
the set FrQ(ab, fe) being closed, we conclude H(p 0 , 0) eFrQ(ab, fs). But since H
is a deformation we have H(p 0 , 0) = p 0 , whence p 0 e Fr Q(ab, !e), which contradicts to p 0 e ab. Thus the inequality t0 >0 is established. _ __
It follows from the definition of t0 that H(abx [0, .t0 ])cQ(ab, -!e). We claim
that there is a number t' e [0, t0 ] such that condition (3) is satisfied. In fact, if t 0 = 1,
then the claim follows from the supposition done in the beginning of the proof.
If t0 < 1, then there exists a sequence of points Pn E ab such that t0 ~ tPn and
t 0 = IimtPn' Let- as previously- {Pnk} be a subsequence of {Pn} converging to
n-+co
a point p 0 e ab. Applying once more the same arguments as above we get
H(p 11k,tPn)eFrQ(ab,fe), whence H(p 0 ,t0 )eFrQ(ab,fe) and thus the point
H(p 0 , t 0 ) is not in ab. Therefore we may take t0 as t'.
Lett' e [0, t 0 ] be any number such that (3) holds and letp 0 E abbe such a point
that H(p 0 , t') is not in ab. Put H(p 0 , t') = q and define
t~ =:= inf{t' e [0, t 0 ]: H{p 0 , t')
= q}.
Thus by the continuity of H we have H(p 0 , t~) = q E X'..ab.
Now let us recall that ab is an R- arc and thus it satisfies condition 3° of
Definition 4. Since p 0 e abc(Limu,u11(a)) n (Limv,vn(a)), there exist two sequences
n-+oo
n-~'oo
2° there is a number e>O such that for almost all positive integers n the sets
u11 b r\ Q(ab, e). and v,a n Q(ab, e) are nonNconnected (thus the points u,(e) and
vtle) are well-defined) and the sets ut, un(e)'.{ U11 (e)} and v,, v,le)'-{v,le)} contain no
ramification point of X;
3° [Lim U11 Un{e)] n [Lim V11 V11(e)] = ab.
231
of points p~ and p~' such that p~ e u,u,(e), p~'
•
Put q~ = H{p~, t~) and q~'
E
v,v11 (e) and Po= limP~= limp~'.
n-too
n-tco
= H(p~', t~) and consider the following two conditions:
(4)
q~ e u,un(e)
for almost all n,
(5)
q~' e v, v,(a)
for almost all n .
232
J. J. Charatonik and Z. Grabowski
Homotopical/y fixed arcs and the contractibility of dendroids
'1
If (4) .holds, then we have
q=
H(p 0 , t~) =lim H(p~, t~) = limq;,_e Lim U11 U 11 (B) ..
II~<Xl
II~<Xl
Similarly if (5) holds, then qe Lim V11 vnCe). Therefore if both (4) and (5) hold, we
have q E (Lim U 11 U11 (e)) n (Lim V11 V 11 (e))cab which contradicts to q E X'-ab.
II~<Xl
/I~<Xl
•
So, either (4) or(5) does not hold. Assume (4) is not true. Thus there is a subsequence {q~J of the sequence {q~} with q~k not in u"kunr,(e). Since the points p;,,, are
in the arcs U11kU11k(e) and q~k are not, and since U11k are end points of the dendroid X,
we conclude by 2° that umJe) e H({p~k} x [0, t~]). Thus there a~·e numbers t,, E [0, t~]
such that H(p~k' t11k) = u11 ~(e). By the compactness of [0, t~] we may assume that the
sequence {t117.} converges to a ·number t" E [0, t~]. Hence it follows from the con~
tinuity "of H that ·
limH(p;k, t11k) = H(p 0 ,
()
eH(abx [0,
t~])<;:H(abx [0, t 0 ])cQ(ab, -!-~.
k->oo
But lim H(p~k' t11k) =lim U11 k(e) which is not in Q(ab,
k~co
k->·oo
-1-i> by the definition of the
points un(e). This contradiction finishes the proof of the theorem.
Proposition 3 and Theorem 5 imply the following
CoROLLARY 6.
If a dendroid
contains an R-are, then it is not contractible.
We shall prove now that all hypotheses of Corollary 6 are essential, i.e., that all
of Definition 4 are necessary to show the non-contractibility of a dendroid
which contains an R-are.
c~nditions
It is easy to verify that H contracts X to its topp (cf. [1], p. 31). Putting, for
the fan X defined above, U 11 = q11 , V 11 = (t, 2 1 - 11) , a= (-!, 0), b = (t, 0) and e =
we see that all the conditions of Definition 4 are satisfied except for v11 are end points
of X. Thus I o is essential.
1
'
i
EXAMPLE 8. Putting,
rn = (2 1 - 11 , 2 1 -n) and
R = PPo
under the same notation
U
io
~ (1, 0),
Pn = (1, 2
q11 = (t, -!·2
-"),
1
-")
for
n=
'n=l
_
[
(1-2t.)e+2t
4(1--t)e
2(1-t) .
7,
U pr2n-1
n=l
co _ _
Upr
n=1
211
_
1,
the top p of which is
the only common point of the both fans. Hence R is contractible.
Further, putting U11 = q211 , vn = 1' 211 - 1 , a= 0;·, 0) and b = p we see that
condition 1° of Definition 4 holds and that for every s such that O<e<-!- the sets
sr
ut,b () Q(ab, 8) are non-connected for almost all n, but vna () Q(ab,
are connected (and thus the points vn(s) cannot be defined). Therefore the first part bfcondition 2° is essential.
To see that the second part is essential consider the following example (due
to P. Mine).
EXAMPLE 9. Let X be the fan described in Example 7, let
and Yn = (t, 2 1 -n) for n = 1, 2, ... and define
X 11
= (!, -!·2 2 -")
M= Xu UxnYn·
, It is easy to observe that M is a contractible dendroid. Taking u, = q1, ,
= (i, 0) and e = { we· see that all conditions mentioned
in Definition 4 are satisfied except that the sets v,v11(e),{v 11 te)} contain ramification
points Yn of M.
= x,, a = (t, 0), b
xy
(we recall that here
stands;for the straight line segment joining points x andy).
So X is a fan with the top p and with end points p 0 and q11 for n = 1, 2, ... DeHne
li: Xx 1-+.X as follows. H(p, t) = p for every t e /, If x ¥: p, then ll(x, t) belongs
to the same ~omponent of the set X"-{P} as tlie point x for every t e I. Further,
if (f! and {! are the radii, i.e. the first polar coordinate~, of the points Ji(x, t) and x
respectively,·. ~e put
-
U
Example
n=1
X= PPo u U (ppn u Pnq,)
e1
P2nq2n)
homeomorphic to X and of a locally connected fan
V11
if
e<! and O~t<t,
if t~e~l and O~t<t,
if
{! < t and t ~ t ~ 1 ,
if l~e~1 and t~t~ 1.
U
in
1, 2, ...
and let
(1 +2t)e
U (PP2n
n=l
as
we see that R is a sub fan of the fan X which can be described as the union of a fan
EXAMPLE 7. Let a pon;.t p be the pole (i.e. tb._e origin) of the polar coordinate
system in .the Euclidean plane. Put in the polar coordinates (e, ({J)
1
233
EXAM~LE 10. Putting, under the sa]ne notations as in Example 7,
s, =
ct, 2
1
-")
and
_
S = PPo u
co__
co _ _
U(PP2n U P2 11 q2n) U U ps2n-1
n=l
n=1
we see that S is a subfan of the fan X which can be described as the union of a fan
homeomorphic to X and of a fan homeomorphic to the harmonic fan, both having
the common top and such that the limit segment of the latter one is contained in the
limit segment of the former. DefineF: Sxl-+Sasfollows.F(p, t)=pforevery tel.
If x e S"'.{p}, then F(x, t) belongs to the same component of the set S".{p} as the
234
J. J. Charatonik and Z. Grabowski
point x for every t e I. Further if lb. and
respectively, we put
Q
are the radii of the points F(x, t) and x
a~t and O~t<f,
if i<rJ<t and O~t<t,
if f~g~1 and O~t<t,
if
rJ~t and t~t~1,
if i<g<t and t~t~1,
if t~rJ~1 and t~t~ 1.
if
{]
(4t+1)g-t
(1-2t)(J+2t
(h = 12(1-t)Q
(1-t)(6g-1)
2(1-t)
In fact. take the fan X described in Example 7 and observe that (6) follows from
the construction and (7) is proved in that place. To define Y put - under the same
notations - Yn = (i, 21 -n) and take
Y = PPo
n--+oO
joining the point ·a with (f, 0) and similarly Lim V11 V11(8) is the straight line segment
IJ-1-0()
joining (}, 0) with the point b. Thus 0 = (Lim U11 U11 (e)) n (Lim vnvnCe))c:.ab and
n--+oo
the opposite inclusion does not hold. So this inclusion is essential.
denote the center of the straight line segment Pn qn; thus W11 =
<2 1 -n. Putting
11
W = PPo u
U (PP2n
n=1
u P2nq2n) u
U(PP2n-1
n=l
G, cp
11 ),
where
U P2n-1 W2n-1)
we see that W is a subfan of the fan X which can be described as the union of two
fans, both homeomorphic to X, having the top p and the limit segment pp0 in common. Further, it is easy to verify that HI Wx J: Wx I-+ W (where His defined in
Example 7) is a contraction of W to its top p. ·Taking un = q211 , v, = w211 _ 1 ,
a = ( t, 0), b = (t, 0) and 8 = i we see that conditions 1° and 2° of Definition 4
are satisfied. Further, we see that Lim U11 U11(e) and Lim V11 V11(e) both are the straight
line segment joining({, 0) with th:-;oint p 0 , whe;c:~he arc ab is a proper subset
of the intersection of these limits: the inclusion abc(Lim u,.u11 (e)) n (Lim v11 v,(e))
n--+oo
n--+oo
does hold but the opposite one does not. So this inclusion is essential.
·Examples 10 and 11 show that every inclusion of equality 3° is essential, so
condition 3° is essential.even in so strong sense.
PROPOSITION
(6)
(7)
(8)
(9)
(10)
12. There exist d-ertdroids X and Y such that
X is a countable plane fan,
X
Y
Y
Y
is
is
is
is
contractible,
contained in X,
not contractible,
contractible in X.
U CPP2n u P2n q2n) u UPY2~-1 ·
n=1
n=l
Thus we see that (8) holds, i.e., Y is a subfan of the fan X, whence it is a plane
fan. Taking u,. = q211 , Vn = Y211 - 1 , a= (f, 0), b = (t, 0) and e =
we see that
all the conditions of Definition 4 are fulfilled with Yfor X, thus ab is an R-are in Y.
Therefore (9) follows from Corollary 6. Further, (7) and (8) imply (10).
It follows from Proposition 12 that Theorem 1 of [3] is not true not only for
dendroids, which was known (see e. g. [5]) but even for plane fans. Thus the following
question seems to be natural:
QUESTION 13. Give an internal characterization of hereditarily contractible
dendroids.
A contribution to the attempt to find such a characterization is the following.
PROPOSITION
EXAMPLE 11. Let, under the same notations as in Example 7, the point w..
i·21 -n< <p
U
i
It is easy to verify that F is a contraction of S to its top p. Putting U11 = q2n,
vn = s2n- 1 , a = (f, 0), b = (-!-, 0) and 8 = t we see that conditions 1° and 2° Defi..
nition 4 hold true. Further, we see that Lim u11 un(8) is the straight line segment
n-~ooo
235
Homotopical/y fixed arcs and the contractibility of dendroids
14.
If a dendroid is smooth, then
it is hereditarily contractible.
Indeed, it is known that the smoothness of dendroids is a hereditary property,
i.e., that every subdendroid of a smooth dendroid is also smooth (see [2], Corollary 6,
p. 299). Further, every smooth dendroid is contractible (see [8], Theorem 1.16,
p. 371; cf. [3], Corollary, p. 93). Thus, for dendroids, the smoothness implies the
hereditary contractibility.
LEMMA 15. If a contractible dendroid c01itairts a point p and a convergent point
sequence {an} such that the sequence of arcs pa11 is convergent, then the limit continuum
Lim pa11 is hereditarily locally connected.
n-+oo
Proof. Let X be a dendroid that satisfies the hypotheses of the lemma, and let
H:· Xxl-+Xbe a contraction of X. Without loss of generality we may assume that
H(Xx {I}) = {p}. Let a= lim a11 • We claim that the point a must pass through all
points of the continuum L:;an during the contraction. In other words we claim
n--+oo
that
(11)
for every point x e Limpa11 there exists a number t e I such that H(a, t) = x.
n-+oo
In fact, for each natural m consider the spherical neighbourhood Q(x, 1/m).
Since x e Limpa11 , there is a natural n0 (m) such that for each n>no(m) we have
pa,. n Q(;,....1im) ::1= 0. Let Xn,m epa11 n Q(x, 1/m) and consider the dendrite
D11 = H({a 11 } xI). Since a,. = H(a 11 , 0) E Dn and p = H(an, 1) E Dn, we have
rn c:. D
whence x n,m e Dn• So there is a number tn •mE I such that H(aM •tn,nJ = x,.,m·
a:rn
n'
Tending with m and n to infinity and considering convergent subsequences If necessary
we get the limit t of {tn,m} with H(a, t) = x. So (11) is proved. .
236
Jlomotopically fixed arcs and the contractibility of dendroids
J. J. Charatonik and Z. Grabowski
It follows from (11) that Limpa11 cH({a}xJ). Since H({a}xJ) is a dendrite,
n-+oo
thus a hereditarily locally connected continuum, therefore the continuum Lim pan is
n-+co
also hereditarily locally connected.
THEOREM
16. If a fan is hereditctrily contJ'actible, then it is smooth.
Pro of. Assume a fan X with the top p is not smooth. Thus there exists a convergent sequence of points x,1 EX such that, putting x = lim X 11 , we have
n-+co
LimpX11'\PX
=;f
0. If the continuum Limpxn is not locally connected, then X is not
n-+co
n-+co
contractible by Lemma 15. So we can assume that Limpx11 is locally connected. Since
n-+co
it is a subcontinuum of the fan X containing the top p, it is also a fan which can reduce
to an arc in some particular case. In any case there is an end point y of Lim px11
which is not in the arc px. By the local connectedness of Limpx11 there is a positive
11-+ co
real number 11 such that
[Lim u,u,(B)] n [Lim V11 V,(e)]
n->w
= {a}. Observe further
237
that the sets unu11 (e) and v11 v,(e)
n-+w
are contained in Q(a, e) and therefore do not contain the only ramification point p
of Y'. So conditions 2° and 3° are fulfilled for the degenerate R-arc{a}. Thus Y' is
not contractible by Corollary 6 which finishes the proof.
Observe that the hypothesis that the dendroid under consideration is a fan is
essential in Theorem 16. Namely, put in the rectangular Cartesian coordinate system
in the plane p = (0, 0), q = (3, 0), p 11 = (1, ljn) and qn = (2, 1/n) for n = 1, 2, ...
and deHne
D = pq u U (PPn u qqn) ·
n=1
It is obvious that D is a non-smooth, hereditarily contractible dendroid having
two ramification points p and q.
Proposition 14 and Theorem 16 imply
CoROLLARY 17. A fan is hereditarily contractible if and only if it is smooth.
ry<tmin{dist(y,px), dist(y, (Limpx11'\PY) u {p})}.
n-+co
·
References
By the definition of an end point of an arc as a point of order 1 (see [7], §51, I,
p. 274) applied to the pointy and to the arc py, there is an open set G such that y E G,
diamG<t17 and 'py n (G".G) is a one-point set. Denote by a the only point of
py n ( G"-G) .. Define (for n = 1, 2, ...) Pn as the first and qn as the last point of the ·
arc px11 ordered from p to X 11 which lies in the boundary G'-.G of G. Taking proper
convergent subsequences if necessary we may assume without loss of generality that
the sequences {Pn} and {q11} are convergent and we s.ee by construction that
limpn = a =lim qn and y E Lim.pnqn. Putting Un = P2n' VII = q~n-1 and
n-+ co
IJ-+co
Y' =
U (pun
INSTITUTJil OF MATHE!lVIATICS OF THE WROCLAW UNIVERSITY
INSTYTUT MATEMATYCZNY UNIWERSYTETU WROCLAWSKIEGO
u pv11 )
11=1
we see that Y' is a subfan of X having U11 and vn as its end points, and that the
sequences {un} and {vn} e1:re convergent to the point a. Thus condition 1° of Definition 4 is satisfied with a = b (the case when the R-are is degenerate). We shall
show that conditions 2° and 3° are satisfied too. Namely, let O<e<imin(ry, d(y, a)),
where d denotes the metric on ·X. Since p E u11 a"-Q(a, e) for almost all1t, the sets
una n Q(a, e) are not connected. Thus un(e) are well defined and- taking convergent
subsequences if necessary- we see that Lim tt11 uu(e)c:.ap by construction. Similarly,
n-+co
since
Y E Limp11 q11"-.Q(a, e)cLim. V11 a"-Q(a, e).
n-+co
n-+co
we see that the sets vna n Q(a, e) are not connected. So vn(e) are well defined and,
using the same arguments as previously we conclude Lhu v11 v11 (e)c:.ay. Therefore
·
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[3] - On contractible denclroids, Colloq. Math. 25 (1972), pp. 89-98.
[4] Sze-Tscn Hu, Homotopy Theory, Academic Press, New York-London 1959.
[5] F. B. Jones, Review of the paper: J. J. Charatonik and C. A. Eberhart (Colloq. Math. 25
(1972), pp. 89-98), Math. Reviews 46, No. 5, # 8193, p. 1412.
[6] K. Kuratowski, Topology I, Academic Press and PWN, New York-London-Warszawa 1966.
[7] - Topology II, Academic Press and PWN, New York-London-Warszawa 1968.
[8] L. Mohler, A characterization of smoothness in dendroids, Fund. Math. 67 (1970), pp. 369-376.
n-+co
Accepte par Ia Redaction le 8. 4. 1976