The Houston Problem Book Problems 101-150

Problems

Problem 101 (HPB): Suppose $M$ is a continuum and $f$ is a monotone mapping of $M$ onto $[0,1]$ such that, if $D$ is a closed proper subset of $M$ which is mapped onto $[0,1]$ by $f$, then $f\big|_D$ is not monotone. Is $M$ irreducible? (Asked by L. Mohler and L.E. Ward 18 October 1978)

Problem 102 (HPB): Is there a non-locally connected continuum $M$ such that there exists a retraction of $2^M$ to $C(M)$? If $M$ is the cone over a compact set with only one limit point, is $C(M)$ a retract of $2^M$? (Asked by J.B. Fugate 18 October 1978)

Problem 103 (HPB): Suppose $M$ is a hereditarily unicoherent and hereditarily decomposable continuum and $f$ is a continuous mapping $M$ into $2^M$. Is there a point $x$ of $M$ such that $x$ is in $f(x)$? (Asked by J.B. Fugate 18 October 1978)

Problem 104 (HPB): Is the open image of a circle-like continuum always circle-like or arc-like? (Asked by J.B. Fugate 18 October 1978)

Problem 105 (HPB): Suppose $M$ is an atriodic $1$-dimensional continuum and $G$ is an upper semi-continuous collection of continua filling up $M$ such that $M/G$ and every element of $G$ are chainable. Is $M$ chainable? (Asked by H. Cook and J.B. Fugate 18 October 1978)

Answer: Comment 1: It follows from a result of Sher that, even if $M$ contains a triod, if $M/G$ and every element of $G$ are tree-like, then $M$ is tree-like. (H. Cook 18 October 1978)

Comment 2: Yes if the requirement that $M$ is $1$-dimensional is replaced by $M$ is strongly unicoherent and $M/G$ is hereditarily decomposable. (W. Dwayne Collins 15 March 1982)

Problem 106 (HPB): If $M$ is tree-like and every proper subcontinuum of $M$ is chainable, is $M$ almost chainable? (Asked by J.B. Fugate 18 October 1978)

Problem 107 (HPB): Suppose $M_1,M_2,\ldots$ is a sequence of mutually disjoint continua in the plane converging to the continuum $M$ homeomorphically. Is $M$ circle-like or chainable? (Asked by J.B. Fugate 18 October 1978)

Problem 108 (HPB): If $M$ is a uniquely arcwise connected continuum, does each light open mapping of $M$ onto itself have a fixed point? (Asked by J.B. Fugate and B. McLean 18 October 1978)

Answer: No (L.G. Oversteegen 22 January 1980)

Problem 109 (HPB): Do pointwise periodic homeomorphisms on tree-like continua have a fixed point? (Asked by J.B. Fugate and B. McLean 18 October 1978)

Problem 110 (HPB): Do disk-like continua have the fixed point property for periodic homeomorphisms? (Asked by J.B. Fugate and B. McLean 18 October 1978)

Problem 111 (HPB): If $M$ is a tree-like continuum with the fixed point property, does $M \times [0,1]$ have the fixed point property? (Asked by J.B. Fugate 18 October 1978)

Problem 112 (HPB): If $M$ is a contractible continuum, do periodic homeomorphisms on $M$ have fixed points? (Asked by J.B. Fugate 18 October 1978)

Problem 113 (HPB): Suppose $M$ is a continuum which is not the sum of a countable monotone collection of proper subcontinua. Is $M$ irreducible about some finite set? (Asked by J.B. Fugate 18 October 1978)

Problem 114 (HPB): Is it true that, if $X$ is a compact metric space, $P$ is a $1$-dimensional connected polyhedron with a geodesic metric and $f$ is an essential mapping of $X$ into $P$, then the span of $f$ is greater than or equal to the span of $P$? (Asked by A. Lelek 6 December 1978)

Answer: No, but a similar and quite strong result in this direction has been recently established by H. Kato, A. Koyama, and E.D. Tymachatyn (A. Lelek, 15 May 1989)

Problem 115 (HPB): Does every atriodic $2$-dimensional continuum contain a $2$-dimensional indecomposable continuum? (Asked by H. Cook 19 February 1979)

Problem 116 (HPB): If $S$ is a compact, uniquely divisible, topological semigroup with $0$, $1$ and no other idempotents, must $S$ have non-zero cancellation? (Asked by D.R. Brown 2 March 1979)

Problem 117 (HPB): Suppose that $M$ is a tree-like continuum such that, for each two points $a,b \in M$, the diagonal in $M \times M$ intersects every continuum containing both $(a,b)$ and $(b,a)$. Does $M$ have span zero? (Asked by H. Cook 2 March 1979)

Problem 118 (HPB): Suppose $H$ and $K$ are two continua with span zero whose intersection is connected and whose sum is atriodic. Does their sum have span zero? (Asked by E. Duda 2 March 1979)

Problem 119 (HPB): Suppose $Y$ is a nondegenerate locally connected continuum and each cyclic element of $Y$ is a completely regular continuum. Is it true that there exsits a continuum $X$ in the plane and a monotone open map of $X$ onto $Y$? (Asked by J. Krasinkiewicz 7 March 1979)

Answer: The answer is yes if $Y$ itself is a completely regular continuum in the plane. (T. Maćkowiak and E.D. Tymchatyn 25 January 1981)

Problem 120 (HPB): Suppose $M$ is the 'canonical' Knaster indecomposable continuum (obtainable by identifying each point of the dyadic solenoid with its inverse) and $h$ is a homeomorphism of $M$ onto itself. Does $h$ have two points fixed? (Asked by W.S. Mahavier 14 March 1979)

Problem 121 (HPB): Does every hereditarily decomposable continuum contain an irreducible continuum with a composant whose complement is degenerate? (Asked by W.S. Mahavier 14 March 1979)

Answer: Yes (L.G. Oversteegen and E.D. Tymchatyn 30 September 1979)

Problem 122 (HPB): Suppose $n$ is a positive integer and $M$ is a continuum such that, for every positive number $\epsilon$, there exists a weakly confluent $\epsilon$-map of $M$ onto an $n$-cell. Does $M$ have the fixed point property? (Asked by H. Cook 15 March 1979)

Answer: Yes for $n=2$. (S.B. Nadler, Fund. Math. 110 (1980), pp.231-232)

Problem 123 (HPB): Does there exist a widely connected complete metric space? (Asked by H. Cook 16 March 1979)

Problem 124 (HPB): Suppose $M$ is the 'canonical' Knaster indecomposable continuum. Does $M$ have a nonseparating closed set that intersects every composant of $M$? (Asked by H. Cook 16 March 1979)

Answer: Yes (W. Debski 8 May 1990)

Problem 125 (HPB): Does there exist a universal hereditarily indecomposable continuum? (Asked by H. Cook 16 March 1979)

Answer: Yes (T. Maćkowiak and P. Minc 24 January 1983)

Problem 126 (HPB): If $M$ is a hereditarily indecomposable continuum containing a pseudo-arc $P$, is $P$ a retract of $M$? (Asked by B. Knaster 16 March 1979)

Answer: Comment 1: J.L. Cornette has shown that each subcontinuum of the pseudo-arc is a retract of it (H. Cook 16 March 1979)

Comment 2: If $M$ is a hereditarily indecomposable continuum, $K$ is a subcontinuum of $M$, and $f$ is a mapping of $K$ into a pseudo-arc $P$, then $f$ can be extended to a mapping of $M$ into $P$. (D. Bellamy 5 May 1979)

Problem 127 (HPB): Does a mapping of the plane into itself with bounded orbits have a fixed point? (Asked by H. Cook 23 March 1979)

Answer: For mappings of the $3$-dimensional Euclidean space, the answer is no. (K. Kuperberg and Coke Reed, Fund. Math., Vol. 114 (1981), p. 229)

Problem 128 (HPB): Given $n>0$, is there a continuous image of the $(2n+1)$-sphere into itself such that the orbit of each point is dense? (Asked by S. Fajtlowicz and D. Mauldin 23 March 1979)

Problem 129 (HPB): Does there exist a chainable continuum $M$ in the plane such that, if $K$ is a chainable continuum in the plane, there exists a homeomorphism $h$ of the plane onto itself that takes $K$ into $M$? (Asked by R.H. Bing 23 March 1979)

Problem 130 (HPB): Is it true that, if $X$ is a continuum, $f$ is a mapping of $X$ into Hilbert space, and $f$ has span zero, then $f$ is almost factoable through $[0,1]$? (Asked by H. Cook, 23 March 1979)

Problem 131 (HPB): Suppose $M$ is a continuum such that, if $G$ is an uncountable collection of nondegenerate subcontinua of $M$, then some two of elements of $G$ have a nondegenerate continuum in their intersection. Does $M$ contain a countable point set that intersects every nondegenerate subcontinuum of $M$? (Asked by H. Cook 26 March 1979)

Problem 132 (HPB): Is there an atriodic tree-like continuum which cannot be embedded in the plane? (Asked by W.T. Ingram 27 March 1979)

Answer: Yes (L.G. Oversteegen and E.D. Tymchatyn 30 September 1981)

Problem 133 (HPB): If $M$ is an atriodic tree-like continuum in the plane, does there exist an uncountable collection of mutually exclusive continua in the plane each memebr of which is homeomorphic? (Asked by R.H. Bing 27 March 1979)

Answer: No (L.G. Oversteegen and E.D. Tymchatyn 30 September 1981)

Problem 134 (HPB): Is there an atriodic tree-like continuum $M$ with positive span which has the property that there exists an uncountable collection $G$ of mutually exclusive continua in the plane such that each member of $G$ is homeomorphic to $M$? (ASked by W.T. Ingram 27 March 1979)

Problem 135 (HPB): Suppose $M$ is a hereditarily indecomposable triod-like continuum such that every proper subcontinuum of $M$ is a pseudo-arc. Can $M$ be embedded in the plane? (Asked by C.E. Burgess 27 March 1979)

Problem 136 (HPB): Given a set $X$ of $n$ points on the plane, a line is "ordinary" if it contains exactly two points of $X$. A point is "ordinary" if it is on two ordinary lines. Is it true that, if not all points of $X$ are on one line, then $X$ contains an ordinary point? (Asked by S. Fajtowicz 28 March 1979)

Answer: Comment 1: No, for $n=6$ (David Jone 16 April 1979)

Comment 2: What is the answer if $n$ is odd? (Asked by S. Fajtlowicz 16 April 1979)

Problem 137 (HPB): Is there a monotonely refinable map from a regular curve of finite order onto a topologically different regular curve of finite order? (Asked by E.E. Grace 11 April 1979)

Problem 138 (HPB): Does there exist, for every $k \leq m \leq \omega$, a space $X$ such that:

(Asked by T.C. Przymusinski 11 April 1979)

Problem 139 (HPB): Does there exist a Lindelöf space $X$ and a complete separable metric space $M$ such that the product space $X \times M$ is not Lindelöf or (equivalently) normal? (Asked by T.C. Przymusinski 11 April 1979)

Problem 140 (HPB): Is a para-Lindelöf space paracompact? (Asked by W.G. Fleissner and G.M. Reed 11 April 1979)

Problem 141 (HPB): Is a collectionwise normal space with a $\sigma$-locally countable base metrizable? (Asked by W.G. Fleissner and G.M. Reed 11 April 1979)

Problem 142 (HPB): Is a sL-cwH, metacompact space para-Lindelöf? (Asked by W.G. Fleissner and G.M. Reed 11 April 1979)

Problem 143 (HPB): Does a para-Lindelöf space with a base of countable order have a $\sigma$-locally countable base? (Asked by W.G. Fleissner and G.M. Reed 11 April 1979)

Problem 144 (HPB): Is there an honest (i.e. provable from ZFC) example of a space of cardinality $\omega_1$ with a $\sigma$-locally countable base which is not perfect? not metacompact? (Asked by W.G. Fleissner and G.M. Reed 11 April 1979)

Problem 145 (HPB): If $X$ has a $\sigma$-disjoint base $B$ and a $\sigma$-locally countable base $B'$, must $X$ have a base $B$ which is simultaneously $\sigma$-disjoint and $\sigma$-locally countable? (Asked by W.G. Fleissner and G.M. Reed 11 April 1979)

Problem 146 (HPB): (1) $a(i)(\mod n(i)), n(1)<n(2)< \ldots < n(k)$, is called covering if every integer satisfies at least one of the congruences (1). Let $c$ ba n arbitrary constant. Is there always a system (1) satisfying $n(1)>c$ which is a covering? (Asked by P. Erdős 18 April 1979)

Answer: $1,000.00 for proof or disproof. Choi has, in Math. of Computation, a system with$n(1)=20$(P. Erdős 18 April 1979) Problem 147 (HPB): Let there be given$n$points in the plane, no$n-k$on a line. Join every two of the points. Prove that they determine at least$a \cdot k \cdot n$distinct lines where$a$is an absolute constant independent of$k$and$n$. (P. Erdős 18 April 1979) Answer: Comment 1:$100.00 for proof or disproof (P. Erdős 18 April 1979)

Comment 2: Proven (J. Beck, Combinatorica 3 (1983), 281-293)

Problem 148 (HPB): Let $G(n)$ be a graph of $n$ vertices. A theorem of Goodman-Posa and myself states that the edges of our graph $G(n)$ can be covered by at most $\dfrac{n^2}{4}$ cliques, (i.e. maximal complete subgraphs). In fact the cliques can be assumed to be edge-disjoint and it suffices to use edges and triangles. Assume now that every edge of $G(n)$ is contained in a triangle. It seems that very much fewer than $\dfrac{n^2}{4}$ cliques will suffice to cover all edges. Determine the value of the least $g(n)$ so that the edges of such a graph can be covered by $g(n)$ cliques. (If too hard then try to determine $\lim \dfrac{g(n)}{n^2}$) (P. Erdős 18 April 1979)

Answer: The number of covering cliques is essentially the same; consider a complete tripartite graph in which two parts are equal and the third one consists of one element. (S. Fajtowicz 1 May 1979)

Problem 149 (HPB): Is it true that the subset $E$ of the complex plane is a type-A (type-B) convergence set if and only if $V_A(E)$ ($V_B(E)$) is bounded? (Asked by F.A. Roach 18 April 1979)

Answer: If 'complex plane' is replaced by 'real line', the resulting statements are true.

Problem 150 (HPB): Let $\mathcal{F}$ be a class of mappings such that all homeomorphisms are in $\mathcal{F}$ and the composition of any two functions in $\mathcal{F}$ is also in $\mathcal{F}$. If $X$ is homogeneous with respect to $\mathcal{F}$, is there a continuum which is $\mathcal{F}$-equivalent to $X$ and which is homogeneous? (Asked by H. Cook 25 April 1979)

Answer: Comment 1: The answer is no for the class $\mathcal{F}$ consisting of all homeomorphisms and all mappings whose range is not homogeneous. (David Jones 9 May 1979)

Comment 2: What is the answer if $\mathcal{F}$ is the class of all mappings? All montone mappings? All open mappings? All finite-to-one mappings? All confluent mappings? (H. Cook 11 May 1979)

Comment 3: No, if $\mathcal{F}$ is the class of all confluent mappings. (H. Kato 9 August 1984)

Comment 4: No, if $\mathcal{F}$ is the class of all mappings; take $X$ to be a harmonic fan as in the paper of P. Krupski in Houston J. Math. 5 (1979), 345-356. (J. Prajs 1 March 1989)