# The Houston Problem Book Problems 51-100

## Problems

Problem 51 (HPB): Is it true that each polyhedron is a the weakly confluent image of a polyhedron whose fundamental group is trivial?

Answer: False, $T2$. (J Grispolakis 19 March 1979)

Problem 52 (HPB): Are strongly regular curves inverse limits of connected graphs with monotone simplicial bonding maps? (Asked by B.B. Epps 19 September 1973)

Answer: No (E.D. Tymchatyn 27 October 1974)

Problem 53 (HPB): If $X$ is the inverse limit of connected graphs with monotone simplicial retractions as bonding maps, is $X$ the weakly confluent image of a dendrite?

Answer: Yes (J. Grispolakis and E.D. Tymchatyn 19 March 1979)

Problem 54 (HPB): Is it true that $X$ is the inverse limit of connected graphs with monotone simplicial retractions as bonding maps if and only if $X$ is locally connected and each cyclic element of $X$ is a graph? (Asked by B.B. Epps, 19 September 1973)

Problem 55 (HPB): Does each continuous mapping of a continuum onto a chainable continuum have property F? (Asked by A. Lelek 26 September 1973)

Answer: No (J.B. Jugate 16 June 1979)

Problem 56 (HPB): Does each weakly confluent mapping of a continuum onto an irreducible continuum have property F? (Asked by A. Lelek 26 September 1973)

Answer: No (J.B. Jugate 16 June 1979)

Problem 57 (HPB): Is it true that each uniformly pathwise connected continuum is uniformly arcwise connected? (Asked by W. Kuperberg 26 September 1973)

Problem 58 (HPB): Suppose $f$ is a continuous mapping of a chainable continuum $X$ onto a nonchainable continuum $Y$. Does there exist a subcontinuum $X'$ of $X$ mapped onto $Y$ under $f$ such that $f \big|_{X'}$ is not weakly confluent? (Asked by A. Lelek 25 January 1981)

Answer: No (T. Maćkowiak and E.D. Tymchatyn 25 January 1981)

Problem 59 (HPB): Does a continuum with zero surjective span have zero span? (Asked by A. Lelek 7 November 1973)

Problem 60 (HPB): Is it true that if $Y=G_1 \cup G_2$ where $G_1$ and $G_2$ are open and $f\big|_{f^{-1}(\overline{G_i})}$ is an MO-mapping for $i=1,2$ then $f$ is an MO-mapping? (Asked by A. Lelek 21 November 1973)

Answer: Spaces are compact metric (A. Lelek 21 November 1973)

Problem 61 (HPB): Suppose $\mathcal{E}$ is the class of all finite one point unions of circles. Is it true that if $X$ and $Y$ are shape-irreducible continua possessing the same shape and there exist curves $C$ and $C'$ in $\mathcal{E}$ such that $X$ is $C$-like and $Y$ is $C'$-like, then there exists a $C$ in $\mathcal{E}$ such that both $X$ and $Y$ are $C$-like? (Asked by A. Lelek 5 December 1973)

Answer: Trivially yes, but there even exists a shape-irreducible figure-eight-like continuum with the shape of a circle that is not circle-like. (J. Segal and S. Spiez 7 May 1986)

Problem 62 (HPB): Are all shapes of polyhedra stable? (Asked by W. Kuperberg 3 April 1974)

Problem 63 (HPB): Given a polyhedron $P$, does there exist a polyhedron $Q$ of the same shape as $P$ and shape stable? (Asked by W. Kuperberg 3 April 1974)

Problem 64 (HPB): Is each uniformly path-connected continuum $g$-contractible? (Asked by D. Bellamy 3 April 1974)

Problem 65 (HPB): Is it true that, for each uniformly path-connected continuum $X$, there exists a compact metric space $Y$ and two mappings $f$, from $X$ onto the cone over $Y$, and $g$, from the cone over $Y$ onto $X$? (Asked by D. Bellamy 3 April 1974)

Answer: No (J. Prajs 1 March 1989)

Problem 66 (HPB): If $C$ is a connected Borel subset of a finitely Suslinean continuum $X$, is $C$ arcwise connected? (Asked by E.D. Tymchatyn 26 September 1974)

Answer: Yes for subsets of regular curves (D.H. Fremlin 12 April 1990)

Problem 67 (HPB): Suppose $X$ is a continuum such that, if $C$ is a subcontinuum of $X$ then the set of all local separating points of $C$ is not the union of countably many closed disjoint proper subsets of $C$. Then, is every connected subset of $X$ arcwise connected? (Asked by E.D. Tymchatyn 26 September 1974)

Problem 68 (HPB): If for each subcontinuum $C$ of the continuum $X$, the set of all local separating points of $C$ is connected, then is every connected subset of $X$ arcwise connected? (Asked by E.D. Tymchatyn 3 October 1974)

Problem 69 (HPB): Do hereditarily indecomposable tree-like continua have the fixed point property? (Asked by B. Knaster 21 November 1974)

Problem 70 (HPB): Do uniquely $\lambda$-connected (or uniquely $\delta$-connected) continua have the fixed point property for homeomorphisms? (Asked by A. Lelek 21 November 1974)

Problem 71 (HPB): Suppose that $X$ is a continuum such that, if $\epsilon > 0$, $C_1,C_2,\ldots$ are mutually separated connected sets of diameter greater than $\epsilon$, and $C_1 \cup C_2 \ldots$ is connected, then there exist mutually exclusive arcs $A_1,A_2,\ldots$ such that, for some subsequence $D_1,D_2,\ldots$ of $C_1,C_2,\ldots$ and each $i$, $A_i$ is a subset of $D_i$, and $\mathrm{diam}(A_i)>\epsilon$. If $X$ locally connected? (Asked by E.D. Tymchatyn 27 January 1975)

Problem 72 (HPB): Suppose $X$ is a continuum such that each $\sigma$-connected subset of $X$ is a semi-continuum. Is $X$ locally connected? (J. Grispolakis 27 January 1975)

Problem 73 (HPB): Suppose $X$ is a continuum such that each $\sigma$-connected $F_\sigma$ subset of $X$ is a semi-continuum. Is $X$ semi-aposyndetic? (Asked by J. Grispolakis 27 January 1975)

Problem 74 (HPB): Let $f$ be a perfect pseudo-confluent mapping of a hereditarily normal, hereditarily locally connected and hereditarily $\sigma$-connected space onto a complete metric space $Y$. Is $Y$ hereditarily $\sigma$-connected? (Asked by J. Grispolakis 3 February 1975)

Answer: The answer is yes if $X$ is also a $(Q=C)$-space. (J. Grispolakis 3 February 1975)

Problem 75 (HPB): Suppose $X$ is a separable metric space which possesses an open basis $\mathscr{B}$ such that, theset $X \setminus G$ is the union of a collection of closed open subsets of $X$. Is $X$ embeddable in a hereditarily locally connected space? (Asked by E.D. Tymchatyn 3 March 1975)

Answer: Yes (L.G. Oversteegen and E.D. Tymchatyn 18 August 1992)

Problem 76 (HPB): Suppose $X$ is a separable metric space such that, if $A$ and $B$ are connected subsets of $X$, then $A \cap B$ is connected. Is it true that if $C$ is a set of non-cut points of $X$, then $X \setminus C$ is connected? (Asked by L.E. Ward 3 March 1975)

Problem 77 (HPB): Is it true that a continuum $X$ is regular if and only if every infinite sequence of mutually disjoint connected subsets of $X$ is a null sequence? (Asked by E.D. Tymchatyn 3 March 1975)

Problem 78 (HPB): Is it true that, for each decomposition of a finitely Suslinean continuum into countably many disjoint connected sets, at least one of them must be rim-compact? (Asked by T. Nishiura 3 March 1975)

Answer: No (T.D. Tymchatyn 30 September 1981)

Problem 79 (HPB): Is it true that no biconnected set with a dispersion point can be embedded into a rational continuum? (Asked by J. Grispolakis 3 March 1975)

Problem 80 (HPB): Is it true that a separable metric space is embeddable into a rational continuum if and only if it posseses an open basis whose elements have scattered boundaries? (Asked by E.D. Tymchatyn 3 March 1975)

Answer: Yes (E.D. Tymchatyn 30 September 1981)

Problem 81 (HPB): Is it true that each continuum of span zero chainable? (Asked by H. Cook and A. Lelek 15 May 1975)

Problem 82 (HPB): Is it true that each continuum of surjective semi-span zero is arc-like? (Asked by A. Lelek 15 May 1975)

Problem 83 (HPB): Suppose $X$ is a connected metric space.

• Is the span of $X$ less than or equal to twice the surjective span of $X$?
• Is the semi-span of $X$ less than or equal to twice the surjective semi-span of $X$?
• Is the surjective semi-span of $X$ less than or equal to twice the surjective span of $X$?
• If $T$ is a simple triod, is the surjective span of $T$ equal to the surjective semi-span of $T$? (Asked by A. Lelek 15 May 1975)

Problem 84 (HPB): Is the confluent image of an arc-like continuum arc-like? (Asked by A. Lelek 21 October 1975)

Problem 85 (HPB): If $f$ is a confluent mapping of an acyclic (or tree-like or arc-like) continuum onto a continuum $Y$, is $f \times f$ confluent? (Asked by A. Lelek 21 October 1975)

Problem 86 (HPB): Do confluent maps of continua preserve span zero? (Asked by H. Cook and A. Lelek 21 October 1975)

Problem 87 (HPB): Is every regularly submetrizable Moore space completely regular? (Asked by H. Cook 13 October 1976)

Problem 88 (HPB): Is every homogeneous continuum bihomogeneous? (Asked by B. Fitzpatrick 27 October 1976)

Answer: No (K. Kuperberg 2 February 1988)

Problem 89 (HPB): Does there exist a noncombinatorial triangulation of the $4$-sphere? (Asked by C.E. Burgess 12 October 1977)

Problem 90 (HPB): Is there a hereditarily equivalent continuum other than an arc or a pseudo-arc? (Asked by H. Cook 2 November 1977)

Problem 91 (HPB): Do hereditarily equivalent continua have span zero? (Asked by H. Cook 2 November 1977)

Problem 92 (HPB): If $M$ is a continuum with positive span such that each of its proper subcontinua has span zero, does every nondegenerate monotone continuous image of $M$ have positive span? (Asked by H. Cook 2 November 1977)

Answer: No (J.F. Davis and W.T. Ingram 30 April 1986; Fund. Math. 131 (1988), 13-24)

Problem 93 (HPB): Does there exist a homogeneous tree-like continuum of positive span? (Asked by W.T. Ingram 8 February 1978)

Answer: Not in the plane (L.G. Oversteegen and E.D. Tymchatyn 23 July 1980)

Problem 94 (HPB): If $M$ is a plane continuum with no weak cut point, is $M$ $\lambda$-connected? (Asked by C.L. Hagopian 12 April 1978)

Answer: Yes, but there exists a counterexample in the $3$-dimensional Euclidean space (C.L. Hagopian 1 April 1979)

Problem 95 (HPB): Is the countable product of $\lambda$-connected continua $\lambda$-connected? (Asked by C.L. Hagopian 12 April 1978)

Answer: Yes, but it is unknown if the product of any two continua is $\lambda$-connected (C.L. Hagopian 7 April 1986)

Problem 96 (HPB): Is the image under a local homeomorphism of a $\delta$-connected continuum also $\delta$-connected? (Asked by C.L. Hagopian 12 April 1978)

Problem 97 (HPB): Can it be proven, without extraordinary logical assumptions, that the plane is not the sum of fewer than $\mathfrak{c}$ mutually exclusive continua? (Asked by H. Cook 19 April 1978)

Answer: The plane is not the sum of fewer than $\mathfrak{c}$ mutually exclusive continua each of which is either Suslinean or locally connected (H. Cook 19 April 1978)

Problem 98 (HPB): Is it true that if $(X,\tau_1)$ is normal, $(X,\tau_2)$ is compact, $\tau_2$ is a subcollection of $\tau_1$, $\mathrm{ind}(X,\tau_1)=0$, and $\mathrm{ind}(X,\tau_2)>0$, then $(X,\tau_1)$ fails to be a Hurewicz space? (Asked by A. Lelek 13 September 1978)

Problem 100 (HPB): Suppose $f$ is a light open mapping from a continuum $M$ onto a continuum $N$, $B$ is a smooth dendroid lying in $N$, and $x$ is a point of $f^{-1}(B)$. Does there exist a smooth dendroid $A$ in $M$ such that $x$ is in $A$ and $f \big|_A$ is a homeomorphism? (Asked by J.B. Fugate 18 October 1978)