The Houston Problem Book Problems 1-50

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Problems

Problem 1 (HPB): Suppose $X$ is a continuum which cannot be embedded in any continuum $Y$ such that $Y$ is the union of a countable family of arcs. Does $X$ contain a connected subset that is not path connected? (Asked by E.D. Tymchatyn, 09/30/81)

Answer: No (E.D. Tymchatyn, 09/30/81).


Problem 2 (HPB): Suppose $X$ is a continuum such that for each positive number $\epsilon$ there are at most finitely many pairwise disjoint connected sets in $X$ of diameter greater than $\epsilon$. Suppose, if $Y$ is any continuous, monotonic, Hausdorff image of $X$, then $Y$ can be embedded in a continuum $Z$ which is the union of a countable family of arcs. Is every connected subset of $X$ path connected?

Answer: Unknown


Problem 3 (HPB): If $X$ is a continuum such that every subcontinuum $C$ of $X$ contains a point that locally locally separates $C$ in $X$, is $X$ regular? (Asked by E.D. Tynchatyn, 09/15/71)

Answer: No (E.D. Tymchatyn, 03/19/79).


Problem 4 (HPB): If $X$ is a Suslinean curve, does there exist a countable set $A$ in $X$ such that $A$ intersects every nondegenerate subcontinuum of $X$? (Asked by A. Lelek, 09/22/71)

Answer: No (P. Minc, 06/15/79).


Problem 5 (HPB): Can every hereditarily decomposable chainable continuum be embedded in the plane in such a way that each end point is accesible from its complement? (Asked by H.Cook, 09/29/71)

Answer: No (P. Minc and W.R.R. Transue, 02/27/90).


Problem 6 (HPB): Is it true that a chainable continuum be embedded in the plane in such a way that every point is accessible from the complement if and only if it is Suslinean? (Asked by H. Cook 09/29/71)

Answer: Yes (P. Minc and W.R.R. Transue, 02/27/90).


Problem 7 (HPB): Is it true that the rim-type of any real curve (or half-ray curve) curve is at most $3$? (Asked by A. Lelek, 09/29/71)

Answer: Yes. A solution follows from Nadler's work. (W. Kuperberg and P. Minc, 03/28/79)


Problem 8 (HPB): Does there exist a rational curve which topologically contains all real curves (or half-ray curves)? (Asked by A. Lelek, 09/29/71)

Answer: Yes (W. Kuperberg and P. Minc, 03/28/79)


Problem 9 (HPB): Does there exist a real curve which can be mapped onto each (weakly chainable continuum) real curve? If not, does there exist a rational curve which can be mapped onto each (weakly chainable) real curve? (Asked by A. Lelek, 09/29/71)

Answer: Delete the parentheses, weakly chainable is needed (W. Kuperberg and P. Minc, 03/28/79)


Problem 10 (HPB): Suppose $D$ is a dendroid, $S$ is compact, finitely generated commutative semigroup of monotone surjections on $D$ which has a fixed end point. Does $S$ have another fixed point? (Asked by L.E. Ward, 20 October 1971)

Answer: Yes, even for $\lambda$-dendroids (W.J. Gray and S. Williams, Bull. Polish Acad. Sci. 27 (1979), 599-604)


Problem 11 (HPB): Is each strongly Hurewicz space an absolute $F_{\sigma}$? (Asked by A. Lelek, 3 November 1971)

Answer: No (E.K. van Douwen, 6 May 1979). The answer appears to be unknown for the metric case (A. Lelek 31 August 1994).


Problem 12 (HPB): Is each product of two Hurewicz metric spaces a Hurewicz space? (Asked by A. Lelek, 3 November 1971)

Answer: No (J.M. O'Farrell 23 March 1984)


Problem 13 (HPB): Is the product of two strongly Hurewicz metric spaces a strongly Hurewicz space? (Asked by H. Cook 3 November 1971)

Answer: Unknown


Problem 14 (HPB): Is every Lindelöf totally paracompact space Hurewicz? (Asked by D.W. Curtis 10 November 1971)

Answer: Unknown


Problem 15 (HPB): Suppose $f$ is an open mapping of a compact metric space $X$ to the Sierpiński curve. Do there exist arbitrarily small closed neighborhoods $U$ of $x$ in $X$ for which $y$ is in $\mathrm{Int}(f(U))$ and $f \big|_U$ is confluent? (Asked by A. Lelek, 24 November 1971)

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


Problem 16 (HPB): Suppose $f$ is a locally confluent and light mapping of a compact metric space $X$ to the Sierpiński curve. Do there exist arbitrarily small closed neighborhoods $U$ of $x$ in $X$ for which $y$ is in $\mathrm{Int}(f(U))$ and $f\big|_U$ is confluent? (Asked by A. Lelek 24 November 1971)

Answer: Yes (T. Maćkowiak and E.D. Tymchatyn 25 January 1981)


Problem 17 (HPB): Is there a topological space $(X,\tau)$ on which the identity is strongly homotopically stable but there is an open mapping onto $X$ which is not strongly homotopically stable? (Asked by H. Cook, 24 November 1971)

Answer: Unknown


Problem 18 (HPB): Is it true that each open mapping of a continuum $X$ onto the Sierpiński curve (or the Menger curve) is strongly homotopically stable at each point of $X$? (Asked by A. Lelek, 24 November 1971)

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


Problem 19 (HPB): Does each strictly non-mutually aposyndetic continuum with no weak cut point contain uncountably many mutually exclusive triods? (Asked by H. Cook 8 December 1971)

Answer?: E.E. Grace has shown that every planar strictly non-mutually aposyndetic continuum has a weak cut point.


Problem 20 (HPB): Suppose $f$ is an open continuous mapping of a continuum $X$ onto $Y$. Does there exist a space $X^*$ such that $X$ is a subset of $X^*$, $X^*$ is a locally connected continuum, and there is an extension $f^*$ of $f$ from $X^*$ to $Y^*$ such that $f^*$ is open and $f^*(X)$ does not intersect $f^*(X^*\setminus x)$? Can $X^*$ be the Hilbert cube? If $X$ is a curve, can $X^*$ be the Menger universal curve? (Asked by A. Lelek 8 December 1971)

Answer: The answer to the first and last question is yes (T. Maćkowiak and E.D. Tymchatyn 25 January 1981)


Problem 21 (HPB): If $X$ is a planar continuum which has only a finite number of complementary domains and $X$ has property A then is $X$ connected im kleinen? (Asked by A. Lelek, 26 January 1972)

Answer: Yes, quite general solution. (T. Maćkowiak and E.D. Tymchatyn, 30 September 1981)


Problem 22 (HPB): Does every totally non-semi-locally connected continuum contain a dense $G_{\delta}$ set of weak cut points? (Asked by E.E. Grave, 2 February 1972)

Answer: Unknown


Problem 23 (HPB): Does every totally non-semi-locally connected bicompact Hausdorff continuum contain a weak cut point? (Asked by E.E. Grace, 2 February 1972)

Answer: Unknown


Problem 24 (HPB): Is it true that if $X$ is a simple tree and the width $w(X) > \epsilon > 0$, then there is a simple triod $T$ in $X$ such that $w(T)>\epsilon$? (Asked by A. Lelek, 9 February 1972)

Answer: No (F.O. McDonald 1 June 1974)


Problem 25 (HPB): Suppose $X$ is a tree-like continuum, its width $w(X)>0$, and $f$ is a continuous mapping from $X$ to a chainable continuum $Y$. Is it true that there exist, for each $a$ such that $0<a<w(X)$, two points $x$ and $y$ such that $d(x,y)=a$ and $f(x)=f(y)$? (Asked by A. Lelek, 9 February 1972)

Answer: Unknown


Problem 26 (HPB): Let $M$ be a locally compact metric space and $a \in M$. Then $a$ is unstable if and only if there exists a homotopy $f(\cdot,t)$ of $M$ to $M$ such that $f(\cdot,0)$ is the identity and $a$ is in $f(M,t)$ for each $t>0$. Also, if and only if there exists a retraction $r$ of $M \times [0,1]$ to $M \times \{0\}$ such that $r^{-1}((a,0))=\{(a,0)\}$. Is local compactness necessary in this theorem? (Asked by W. Kuperberg, 13 September 1972)

Answer: Unknown


Problem 27 (HPB): Suppose $M$ is an $n$-dimensional compact metric space and $A$ is the set of all unstable points of $M$. Does there exist a homotopy $f(\cdot,t)$ of $M$ to $M$ such that $f(\cdot,0)$ is the identity and $f(M,t)$ and $A$ do not intersect for each $t>0$? (Asked by W. Kuperberg, 13 September 1972)

Answer: No (W. Kuperberg and P. Minc, 28 March 1979)


Problem 28 (HPB): Prove that the Topologist's sine curve is not pseudo-contractible. (Asked by W. Kuperberg, 20 September 1972)

Answer: Done (W. Debski, 8 May 1990)


Problem 29 (HPB): Does there exist a $1$-dimensional continuum which is pseudo-contractible but is not contractible? (Asked by W. Kuperberg, 20 September 1972)

Answer: Unknown


Problem 30 (HPB): Suppose $X$ and $Y$ are continua (or even polyhedra), $x \in X$, $y \in Y$, and there exist neighborhoods $U$ and $V$ of $x$ and $y$ in $X$ and $Y$, respectively, such that a homeomorphism of $U$ onto $V$ transforms $x$ onto $y$. Is it true that if $x$ is pseudo-unstable in $X$, then $Y$ is pseudo-unstable in $Y$? (Asked by W. Kuperberg, 20 September 1972)

Answer: Unknown


Problem 31 (HPB): Is it true that the pseudo-arc is not pseudo-contractible? (Asked by W. Kuperberg, 20 September 1972)

Answer: Unknown


Problem 32 (HPB): Does there exist a mapping $f$ of the circle or the plane onto itself such that $f^n$ has a fixed point for each $n \geq 2$, but $f$ does not have a fixed point? (Asked by W. Kuperberg, 11 October 1972)

Answer: Yes; L. Block, Trans. Amer. Math. Soc. 260 (1980), 553-562, and, independentl, M. Cook, oral communication, 1988. (W.T. Ingram, 30 June 1989)


Problem 33 (HPB): Suppose $P$ is a subset of the positive integers with the property that, if $k \in P$, the $n \cdot k$ is in $P$ for each positive integer $n$. Does there exist a locally connected continuum $X$ and a mapping $f$ of $X$ onto $X$ such that $f^n$ has a fixed point if and only if $n \in P$? (Asked by W. Kuperberg, 11 October 1972)

Answer: Yes (M. COok, 20 February 1989)


Problem 34 (HPB): Is it true that if $f$ is a mapping of a tree-like continuum into itself, then there exists an $n$ such that $f^n$ has a fixed point? (Asked by W.T. Ingram, 11 October 1972)

Answer: No (P. Minc, 15 June 1991)


Problem 35 (HPB): Suppose $f$ is a continuous mapping of a continuum $X$ onto a continuum $Y$, $Y=H \cup K$ is a decomposition of $Y$ into subcontinua $H$ and $K$, $f \big|_{f^{-1}(H)}$ and $f\big|_{f^{-1}(K)}$ are confluent, and $H \cap K$ is a continuum which does not cut $Y$ and is an end continuum of both $H$ and $K$. Is $f$ confluent? (Asked by W.T. Ingram, 11 October 1972)

Answer: Unknown


Problem 36 (HPB): Suppose $f$ is a weakly confluent and locally confluent mapping of a continuum $X$ onto a tree-like continuum $Y$. Does there exist a subcontinuum $L$ of $X$ such that $f(L)=Y$ and $f\big|_L$ is confluent? (Asked by D.R. Read, 25 October 1972)

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


Problem 37 (HPB): Suppose $(X, \leq)$ is a finite partially ordered set. By taking $\overline{\{y\}}=\{x \colon x \leq y \}$ one gets a $T_0$ topology on $X$; $\overline{A}$ is the union of the closures of the points of $A$ and then a mapping $f$ of $X$ into $X$ is continuous if and only if $f$ is order preserving. A retraction $r$ of $X$ onto $Y$ is "relating" provided that $x \in X \setminus Y$ implies $x$ and $r(x)$ are related. Does there exist a unique minimal relating retract of $X$? (Asked by L.E. Ward, 29 November 1972)

Answer: Unknown


Problem 38 (HPB): Suppose $X$ in HPB Problem 37 is $A \cup B$, where $A$ and $B$ are closed sets and $A,B$, and $A \cap B$ have the fixed point property. Does $X$ have the fixed point property? (L.E. Ward, 29 November 1972)

Answer: Unknown


Problem 39 (HPB): Suppose $X$ is a finite partially ordered set with the fixed point property. Is $X$ unicoherent? (Asked by L.E. Ward 29 November 1972)

Answer: Unknown


Problem 40 (HPB): Does there exist a nonseparating (locally compact) subset of the plane which is neither compact nor open in the plane such that $f$ is a continuous and reversible function of $X$ onto a subset of the plane, then $f$ is a homeomorphism? (Asked by A. Lelek, 7 February 1973)

Answer: Unknown


Problem 41 (HPB): Suppose that the plane is the union of two disjoint sets $A$ and $B$ neither of which contains a Cantor set. Is each continuous and one-to-one function of $A$ into the plane a homeomorphism? (Asked by A. Lelek 14 July 1980)

Answer: No (E.K. van Douwen 14 July 1980)


Problem 42 (HPB): Suppose $X$ is a compact metric space and $S$ is the set of all stable points of $X$. Suppose $\mathrm{dim} \hspace{2pt} X$ is finite and $a$ is an infinite cycle in $S$. Is it true that if $a \sim 0$ in $X$, then $a \sim 0$ in $S$? (Asked by T. Ganea, 7 March 1973)

Answer: Yes, solved by Namioka. (W. Kuperberg and P. Minc 28 March 1979)


Problem 43 (HPB): Suppose $X$ is a curve that is contractible, or pseudo-contractible, or contractible to a proper subset only. Does there exist in $X$ a point which is unstable (or pseudo-unstable, respectively, or otherwise)? (Asked by W. Kuperberg, 7 March 1973)

Answer: Unknown


Problem 44 (HPB): Suppose $X$ is a metric space; does superstable in $X$ imply 'stable'? (Asked by W. Kuperberg 7 March 1973)

Answer: Unknown


Problem 45 (HPB): It is true that if $Y$ is an arcwise connected continuum such that each continuous mapping of a continuum onto $Y$ is weakly confluent, then $Y$ is an arc? (Asked by A. Lelek, 21 March 1973)

Answer: Yes, since $Y$ is irreducible, unicoherent, and not a triod (J. Grispolakis, 19 March 1979)


Problem 46 (HPB): Suppose $Y$ is a compact metric space such that the Borsuk modified fundamental group of $Y$ is $0$ and $f$ is a local homeomorphism of $X$ onto $Y$. Must $f$ be a homeomorphism? (Asked by W. Kuperberg 11 April 1973)

Answer: Unknown


Problem 47 (HPB): Do atomic mappings preserve semi-aposyndesis (or mutual aposyndesis) of continua? (Asked by W.T. Ingram, 18 April 1973)

Answer: Yes, such mappings must either be homeomorphisms or constant mappings (T. Maćkowiak 30 September 1981)


Problem 48 (HPB): Is each unicoherent and mutually aposyndetic continuum locally connected? (Asked by L. Gibson 18 April 1973)

Answer: No, but the answer is unknown for such continua which are $1$-dimensional (T. Maćkowiak 30 September 1981)


Problem 49 (HPB): Supppose $f$ is an open mapping of a continuum $X$ onto $Y$ and $y$ is a branch-point of $Y$. Is there a branch-point of $X$ such that $f(x)=y$? (Asked by V. Parr, 25 April 1973)

Answer: No, but the answer is yes if $f$ is light (T. Maćkowiak 30 September 1981)


Problem 50 (HPB): Suppose $f$ is a weakly confluent mapping of a continuum $X$ onto $Y$ and $y$ is a branch-point of $Y$. Is there a branch-point $x$ of $X$ such that $f(x)=y$? (Asked by V. Parr 25 April 1973)

Answer: No (T. Maćkowiak 30 September 1981)