The Houston Problem Book Problems 201-218

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Problems

Problem 201 (HPB): If $C$ is a simple closed curve, is the span of $C$ equal to the semi-span of $C$? (Asked by A. Lelek 16 January 1989)

Answer: An affirmative answer to part (4) of Problem 83 would establish an analogous result for simple triods. (A. Lelek 16 January 1989)


Problem 202 (HPB): Is each separable metric, hereditarily locally connected space of dimension less than or equal to $1$? (Asked by E.D. Tymchatyn 27 February 1989)

Answer: Unknown


Problem 203 (HPB): Can the space of homeomorphisms of the Menger universal curve be embedded as the set of end points of an $\mathbb{R}$-tree? (Asked by E.D. Tymchatyn 27 February 1989)

Answer: Yes (K. Kawamura, L.G. Oversteegen and E.D. Tymchatyn 15 October 1993)


Problem 204 (HPB): For separable metric spaces, does condition R1 imply condition R2? (Asked by L.G. Oversteegen and E.D. Tymchatyn 14 April 1989)

Answer: Comment 1: If the answer to Problem 204 is yes, then the answer to Problem 197 is no. (E.D. Tymchatyn 14 April 1989)

Comment 2: Yes (L.G. Oversteegen and E.D. Tymchatyn 18 August 1992)


Problem 205 (HPB): Suppose $X$ is a separable metric space and there exists a countable subset $C$ of $X$ such that $X \setminus C$ satisfies condition R2. Is $\mathrm{dim}(X) \leq 1$? (Asked by E.D. Tymchatyn 14 April 1989)

Answer: An affirmative solution of Problem 205 would imply an affirmative solution of Problem 202 (E.D. Tymchatyn 14 April 1989)


Problem 206 (HPB): Are the sets of all end points and of all branch points (in the classical sense) of each dendroid Borel?(Analytic?) (Asked by A. Lelek and J. Nikiel 21 April 1989)

Answer: Comment 1: The fact that the set of all end points of each planar dendroid is $G_{\delta \sigma\delta}$ has been established in Fund. Math. 49 (1961), pp. 301-319. (A. Lelek 21 April 1989)

Comment 2: No, neither set need be Borel. However, the set of end points is co-analytic and hte set of branch-points is analytic. (J. Nikiel and E.D. Tymchatyn 6 February 1990)


Problem 207 (HPB): Can each $n$-dimensional, where $n>1$, locally connected (metric) continuum be represented as the inverse limit of a sequence of $n$-dimensional polyhedra with monotone surjections as bonding maps? (Asked by J. Nikiel 21 April 1989)

Answer: If $n=1$, the answer is no. (J. Nikiel 21 April 1989)


Problem 208 (HPB): Suppose $X$ is the inverse limit of a sequence of Hausdorff continua such that each of them is a continuous image of a Hausdorff arc and the bonding maps are monotone surjections. Is $X$ also a continuous image of a Hausdorff arc? (Asked by J. Nikiel 21 April 1989)

Answer: Yes (J. Nikiel, H.M. Tuncali and E.D. Tymchatyn 29 June 1991)


Problem 209 (HPB): Let $X$ be a Hausdorff space such that $X$ is a continuous image of an orderable, compact Hausdorff space. Is $X$ supercompact? (Regular supercompact?) (Asked by J. Nikiel 21 April 1989)

Answer: Comment 1: It is known that if $X$ is assumed, in addition, to be zero-dimensional or metrizable, then $X$ is regular supercompact. A less general question was posed in 1977 by J. van Mill; he asked if each rim-finite continuum is supercompact. (J. Nikiel 21 April 1989)

Comment 2: Yes (W. Bula, J. Nikiel, H.M. Tuncali and E.D. Tymchatyn 15 November 1990)


Problem 210 (HPB): Suppose $X$ is a monotically normal, compact space

Answer: It is known that continuous images of orderable, compact Hausdorff spaces are monotically normal. Moreover, P. Nyikos and S. Purisch proved in 1987 that monotonically normal, scattered, compact spaces are continuous images of well-ordered, compact Hausdroff spaces. Part (1) is related to the following problem raised in 1973 by S. Purisch: Is each monotonically normal, separable, zero-dimensional, compact space always orderable? (J. Nikiel 21 April 1989)


Problem 211 (HPB): Is each monotonically normal, compact space a continuous image of a monotonically normal, zero-dimensional, compact space? (Asked by J. Nikiel 21 April 1989)

Answer: Unknown


Problem 212 (HPB): Suppose $X$ is a monotonically normal, separable, zero-dimensional, compact space. Is it true that if $\mathcal{C}$ is a collection of mutually disjoint closed subsets of $X$ such that $\mathcal{C}$ is a null-family, then the collection of sets belonging to $\mathcal{C}$ whic have at least $3$ elements is countable? (Asked by J. Nikiel 21 April 1989)

Answer: Unknown


Problem 213 (HPB): Let $X$ be a dendron. Does there exist a hereditarily indecomposable Hausdorff continuum $Y$ such that $X$ can be embedded in the hyperspace $C(Y)$ of subcontinua of $Y$? (Asked by J. Nikiel 21 April 1989)

Answer: If $X$ is a metrizable dendron, then $X$ can be embedded in $C(Y)$ for each hereditarily indecomposable metric continuum $Y$. (J. Nikiel 21 April 1989)


Problem 214 (HPB): Does there exist a universal totally regular continuum? (Asked by J. Nikiel 21 April 1989)

Answer: Yes (J. Buskirk 11 October 1991)


Problem 215 (HPB): Does there exist a continuous function $f$ defined on the closed unit interval with values in a metric space such that $f$ is at most $n$-to-one, for some positive integer $n$, and $f$ is note finitely linear? (Asked by A. Lelek 15 May 1989)

Answer: Unknown


Problem 216 (HPB): Does there exist a monotone retraction $r \colon D \rightarrow C$ of the disk $D$ onto its boundary $C$ (that is, $r^{-1}(y)$ connected and $r(y)=y$ for each $y \in C$, and $r$ not necessarily continuous) such that all but a finite number of points of $C$ are values of continuity of $r$? (Asked by A. Lelek 15 May 1989)

Answer: Unknown


Problem 217 (HPB): Does there exist, for every integer $n>2$, a finite-dimensional separable metric space $X$ and its finite-dimensional metric compactification $cX$ such that if $dX$ is a metric compactification of $X$ and $cX$ follows $dX$, then $\mathrm{dim}(dX) \geq n+ \mathrm{dim}(X)$? (Asked by A. Lelek 15 May 1989)

Answer: The answer is yes for $n=2$. (A. Lelek 15 May 1989)


Problem 218 (HPB): Let $G$ be the $3$-dimensional unit cube in the Euclidean $3$-space $\mathbb{R}^3$, and let $A$ be a countable union of planes contained in $\mathbb{R}^3$ (not necessarily parallel). Suppose $f$ is a continuous mapping of $G$ into a metric space $Y$ such that if $f^{-1}(y)$ is nondegenerate, for a point $y \in Y$, then $f^{-1}(y)$ is contained in $A$. Is $\mathrm{dim}(f(G)) \geq 3$? (Asked by A. Lelek 14 May 1989)

Answer: Comment 1: An affirmative answer to Problem 218 implies an affirmative answer to Problem 217 for $n=3$. The relationship between these problems and some other similar compactification problems was described in the proceedings of a symposium: Contributions to Extension Theory of Topological Structures, Berlin 1969, pp. 147-148. The importance of compactifications is not undermined by the fact that they have been only sporadically discussed at the topology seminar of whose $18$ year this book is record, more or less. (A. Lelek 15 May 1989)