# The Houston Problem Book Problems 151-200

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## Problems

Problem 151 (HPB): Suppose, in the previous problem, $X$ is a homogeneous continuum, $Y$ is a continuum which is $\mathcal{F}$-equivalent to $X$, and $\epsilon$ is a positive number. Does there exist a positive number $\delta$ such that if $a$ and $b$ are two points of $Y$ such that the distance from $a$ to $b$ is less than $\delta$, then there is a mapping $f$ in $\mathcal{F}$ from $Y$ onto $Y$ such that $f(a)=b$ and no point of $Y$ moved a distance more than $\epsilon$? (Asked by H. Cook 25 April 1979)

Answer: If the answer to Problem 150 is 'yes', then the answer to this problem is 'no'. (H. Cook 25 April 1979)

Problem 152 (HPB): Let $M_n$ be the lattice consisting of $0,1,$ and $n$ mutually complementary elements and $P_m$, the lattice of all partitions of an $m$-element set. Let $f$ be a function (one-to-one function) of $M_n$ into $P_m$. What is the probability that $f$ is a homomorphism (an automorphism)? (Asked by S. Fajtlowicz 20 May 1979)

Problem 153 (HPB): Suppose $M$ is a continuum and $f$ is a mapping of $M$ onto itself such that, for each point $x$, $M$ is irreducible from $x$ to $f(x)$. Is there an essential mapping of $M$ onto a circle? (Asked by H. Cook 6 June 1979)

Problem 154 (HPB): Is the space of automorphisms of the pseudo-arc connected? (Asked by J> Krasinkiewicz 14 November 1979)

Problem 155 (HPB): Are planar dendroids weakly chainable? (Asked by J. Krasinkiewicz 14 November 1979)

Problem 156 (HPB): Let $X$ is a nondegenerate homogeneous hereditarily decomposable continuum. Is it true that $X$ is homeomorphic to a circle? (Asked by J. Krasinkiewicz 14 November 1979)

Problem 157 (HPB): Does there exist a finite (countable) to one mapping from a hereditarily indecomposable continuum onto a hereditarily decomposable continuum? (Asked by J. Krasinkiewicz 14 November 1979)

Problem 158 (HPB): Let $X$ be a nondegenerate continuum such that there exists a continuous decomposition of the plane into elements homeomorphic to $X$. Must $X$ be the pseudo-arc? (Asked by J. Krasinkiewicz 14 November 1979)

Problem 159 (HPB): Does there exist a dendroid $D$ such that the set $E$ of end points is closed and each point of $D \setminus E$ is a ramification point? (Asked by J. Krasinkiewicz 14 November 1979)

Answer: Yes (J. Nikiel 1 January 1984)

Problem 160 (HPB): Let $X$ be a continuum such that the cone over $X$ embeds in Euclidean $3$-space. Does $X$ embed in the $2$-sphere? (Asked by J. Krasinkiewicz 14 November 1979)

Problem 161 (HPB): If the set function $T$ is continuous for the Hausdorff continuum $S$, is it true that $S$ is $T$-additive? (Asked by D. Bellamy 18 February 1980)

Answer: A bushel of Extra Fancy Stayman Winesap apples for a solution. (D. Bellamy 21 Februray 1980)

Problem 162 (HPB): If $T$ is continuous for the (Hausdorff) continuum $S$, is it true that the collection $\{T(p) \colon p \in S\}$ is a continuous decomposition of $S$ such that the quotient space is locally connected? (Asked by D. Bellamy 18 Februray 1980)

Problem 163 (HPB): If $T$ is continuous for $S$ and there is a point $p \in S$ such that $T(p)$ has nonempty interior, is $S$ indecomposable? (Asked by D. Bellamy 18 Februray 1980)

Problem 164 (HPB): If $X$ and $Y$ are indecomposable continua, is $T$ idempotent on $X \times Y$? Even for only the closed sets in $X \times Y$? (Asked by D. Bellamy 18 February 1980)

Problem 165 (HPB): If $X$ and $Y$ are indecomposable continua, $S=X \times Y$, and $W$ is a subcontinuum of $S$ with nonempty interior, is $T(W)=S$? (Asked by F.B. Jones 18 Februrary 1980)

Problem 166 (HPB): If $X$ is a compact metric continuum, $p \in X$, $Y=X \setminus \{p\}$, and $\beta(Y) \setminus Y$ is an indecomposable continuum, must $X$ be locally connected (connected im kleinen) at $p$? Is it true that $X = M \cup K$ where $M$ is compact, $p \not\in M$, and $K$ is irreducible from some point $q$ to $p$? (Asked by D. Bellamy 18 Februray 1980)

Answer: Comment: $\beta(X)$ is the Stone–Čech compactification of $X$.

Problem 167 (HPB): Suppose $M$ is a noncompact $n$-manifold for some $n>1$ and $M$ has no two-point compactification. Is $\beta(M)\setminus M$ necessarily an aposyndetic continuum? (Asked by D. Bellamy 18 February 1980)

Answer: This is true if $M$ is a Euclidean $n$-space.

Problem 168 (HPB): Does there exist a thin space with infinitely many points? (Asked by P.H. Doyle 18 Februray 1980)

Answer: The only known examples of thin spaces are finite spaces which are the product of a discrete space with an indiscrete space.

Problem 169 (HPB): Is every aposyndetic, homogeneous, $1$-dimensional continuum locally connected? (Asked by J.T. Rogers 26 April 1982)

Answer: No (J.T. Rogers 26 April 1982)

Problem 170 (HPB): Do there exist even integer $i$ and $j$ such that, for every odd integer $k$ and sequence $A_1,A_2,A_3,\ldots$ such that $A_1=k$ and, for each $n$, $A_{n+1}\setminus A_n$ is either $i$ or $j$, some term of that sequence is prime? (Asked by J.T. Lloyd 14 November 1980)

Answer: If the prime $k$-tuple conjecture of Hardy and Littlewood is true, then $A_n$ is a prime infinitely often for any $i$ and $j$. In case $i=2$ and $j=4$, infinitely many prime twins suffice. (P. Erdős 24 November 1981)

Problem 171 (HPB): Is every weakly chainable atriodic tree-like continuum chainable? (Asked by Lee Mohler 16 April 1981)

Problem 172 (HPB): Does there exist a continuum $M$ such that no monotone continuous image of $M$ contains a chainable continuum? (Asked by H. Cook 13 May 1981)

Problem 173 (HPB): Do there exist, in the plane, two simple closed curves $J$ and $C$ such that $C$ is in the bounded complementary domain of $J$ and the span of $C$ is greater than the span of $J$? (Asked by H. Cook 15 May 1981)

Problem 174 (HPB): In a topological space $(X,\tau)$, each set $A$ has its derived set, $$A^d = \{x \in X \colon x \in \overline{A \setminus \{x\}} \}.$$ Does there exist a $T_0$-space $(x,\tau)$ and a subset $A$ of $X$ such that the second derived set $(A^d)^d$ is not closed in $X$? (Asked by A. Lelek 15 May 1981)

Problem 175 (HPB): In how many ways can one put $k$ dominoes on an $n \times n$ chessboard so that each covers exactly two squares? (Asked by S. Fajtlowicz 15 May 1981)

Answer: The answer is unknown for $k<3$. (S. Fajtlowicz 12 March 1986)

Problem 176 (HPB): Suppose $X_1,x_2,X_3,\ldots$ is a sequence of positive numbers which converges to zero. Is there a set of positive measure which does not contain a set similar to $\{X_1,X_2,X_3,\ldots\}$? (Asked by P. Erdős 24 November 1981)

Problem 177 (HPB): The equation $n!=m!k! (m>k)$ is solved for $n=10$, $m=7$, $k=6$, and also (trivially) if $n=k!, m=k!-1$. Are there any other solutions? (Asked by D. Levine 13 July 1982)

Problem 178 (HPB): Does there exist, for each $t \in \left( \dfrac{1}{2}, 1 \right)$, a simple triod $X(t)$ on the plane such that, with the natural metric, we have $\dfrac{\sigma^*\left(X(t)\right)}{\sigma\left(X(t)\right)}=t?$

Answer: Comment: $\sigma^*\left(X(t)\right)$ and $\sigma\left(X(t)\right)$ denote, respectively, the surjective span and the span of $X(t)$.

Problem 179 (HPB): Is there a tree-like continuum $X$ on the plane such that $X$ is weakly chainable but a subcontinuum of $X$ is not? (Asked by P. Minc 24 January 1988)

Problem 180 (HPB): Suppose $X$ is an arcwise connected continuum with a free arc and such that, for each $p \in X$, there exists a homeomorphism $h \colon X \rightarrow X$ with $h(x)=x$ if and only if $x=p$. Is $X$ a simple closed curve? (Asked by H. Cook 31 October 1988)

Answer: No (H. Gladdines 15 June 1994)

Problem 181 (HPB): Suppose $X$ is an arcwise connected continuum with a free arc and with the fixed set property for monotone onto maps. Is $X$ a simple closed curve? (Asked by Y. Ohsuda 31 October 1983)

Problem 182 (HPB): Is it true that if $T$ and $T'$ are trees with $T'$ contained in $T$, then the surjective span of $T$ is greater than or equal to one-half width of $T'$? What about continua $T$ and $T'$ with $T'$ contained in $T$? (Asked by A. Lelek 12 October 1984)

Problem 183 (HPB): Let $R$ be a space having a topological property $(P)$. Is there an $R$-monolithic (locally $R$-monolithic) space with property (P), where (P) is one of the following properties: countable, second countable, Moore? (Asked by S. Iliadis 8 February 1985)

Problem 184 (HPB): Does an open mapping preserve span zero of continua? (Asked by E. Duda 22 February 1985)

Answer: Yes (K. Kawamura 16 March 1987)

Problem 185 (HPB): Is the product of two unicoherent continua always unicoherent? (Asked by E. DUda 22 February 1985)

Answer: No (A. García-Máynez and A. Illanes 15 June 1989)

Problem 186 (HPB): Characterize mappings $f \colon X \twoheadrightarrow Y$ such that if $H$ is a proper subcontinuum of $Y$, then there is a proper subcontinuum $K$ of $X$ such that $f(K)$ contains $H$. (Asked by E. Duda 22 February 1985)

Problem 187 (HPB): Does the class of approximable mappings coincide with that of the confluent ones? (Asked by W. Debski 22 March 1985)

Problem 188 (HPB): In each continuum of span zero continuously ray extendable? (Asked by W.T. Ingram 14 October 1985)

Problem 189 (HPB): Does every strongly infinite-dimensional absolute $G_{\delta}$ separable metric space contain a $G_{\delta}$ subspace which is totally disconnected and hereditarily strongly infinitely-dimensional? (Asked by A. Lelek 4 November 1985)

Problem 190 (HPB): Is it true that if $A$ is a subset of the Euclidean $n$-space $\mathbb{R}^n$ and $\mathrm{dim}(A) \leq n-2$, then every two points in $\mathbb{R}^n \setminus A$ can be joined by a $1$-dimensional continuum contained in $\mathbb{R}^n \setminus A$? (Asked by J. Krasinkiewicz 16 May 1986)

Problem 191 (HPB): Is it true that if $A$ is a subset of $\mathbb{R}^n$, then there exists a subset $B$ of $\mathbb{R}^n$ which contains $A$ and is such that $\mathrm{dim}(B)=\mathrm{dim}(A)$ and $\mathrm{dim}\left( \mathbb{R}^n \setminus B \right)=n-\mathrm{dim}(B)-1$? (Asked by J. Krasinkiewicz 16 May 1986)

Answer: An affirmative solution of Problem 191 would imply an affirmative solution of Problem 190. (J. Krasinkiewicz 16 May 1986)

Problem 192 (HPB): Does there exist a hereditarily infinite-dimensional metric continuum with trivial shape? (Asked by J. Krasinkiewicz 12 June 1986)

Problem 193 (HPB): Is it true that if $X$ is a homogeneous finite-dimensional metric nondegenerate continuum with trivial shape, then $X$ is $1$-dimensional? (Asked by J. Krasinkiewicz 12 June 1986)

Problem 194 (HPB): Let $X$ be a homogeneous curve such that the rank, $r$, of the first Čech cohomology group of $X$ with integer coefficients is finite. Is $r \leq 1$? (Asked by J. Krasinkiewicz 12 June 1986)

Problem 195 (HPB): Let $X_1,X_2,\ldots$ be an inverse sequence of polyhedra with bonding maps $P_n \colon X_{n+1} \rightarrow X_n$ such that the inverse limit is a hereditarily indecomposable continuum. Let $F_n$ be a continuous mapping of the $2$-sphere into $X_n$ such that $F_n$ is homotopic to the composite $P_n[F_{n+1}]$ for $n=1,2,\ldots$. Is $F_1$ homotopic to a constant mapping? (Asked by J. Krasinkiewicz 12 June 1986)

Problem 196 (HPB): Does there exist a piecewise linear mapping $f$ of a tree onto itself such that $\sigma(f^2)>0$ and $\displaystyle\lim_{n \rightarrow \infty} \sigma(f^n)=0$? (Asked by W.T. Ingram 10 November 1986)

Answer: Notation: $\sigma(f^n)$ denotes the span of $f^n$

Comment: Yes. In fact, for each positive integer $n$ there is a piecewise linear mapping of a triod onto itself so that $\sigma(f^i)>0$ for each $i \leq n$ and $\sigma(f^i)=0$ for $i >n$. (S.W. Young 2 November 1989)

Problem 197 (HPB): Does there exist a separable metric space $X$ whose dimension is greater than $1$ and such that $X$ satisfies condition R1? (Asked by L.G. Oversteegen 13 November 1987)

Answer: No (L.G. Oversteegen and E.D. Tymchatyn 18 August 1992)

Problem 198 (HPB): Is the property of being weakly chainable a Whitney property? (Asked by H. Kato 14 November 1988)

Problem 199 (HPB): If $X$ is a tree-like continuum (or a dendroid), does $C(X)$ have the fixed point property? (Asked by H. Kato 14 November 1988)
Problem 200 (HPB): Is it true that there do not exist expansive homeomorphisms on any Peano curve (that is, a locally connected $1$-dimensional continuum, in particular, the Menger universal curve)? (Asked by H. Kato 21 1988)