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{ } XYYXX Ε X Ε YXYZXXCXCXX P LLUL Pxyp UU Pxy XAXAX UNIVERSITY OF HOUSTON MATHEMATICS PROBLEM BOOK (10/14/94) Problem 1. SupposeX is a continuum which cannot be embedded in any contin- uumYY such that is the union of a countable family of arcs. Does X contain a connected subset that is not path-connected? (E. D. Tymchatyn, 09/15/71) Comment: The answer is no. (E. D. Tymchatyn, 09/30/81) Problem 2. SupposeX is a continuum such that for each positive number there are at most finitely many pairwise disjoint connected sets inX of diameter greater thanY . Suppose, if is any continuous, monotonic, Hausdorff image of XY , then can be embedded in a continuumZ which is the union of a countable family of arcs. Is every connected subset ofX path-connected? (E. D. Tymchatyn, 09/15/71) Problem 3. IfXCX is a continuum such that every subcontinuum of contains a point that locally separatesCX in , is X regular? (E. D. Tymchatyn, 09/15/71) Definition. A pointpL of a locally compact separable metric space is a local separating point ofLULp provided there exists an open set of containing and two pointsxy and of the component containing pU of such that Up\{ } is the sum of two mutually separated point sets, one containingxy and the other containing . Definition. A continuum isregular if each two of its points can be separated by a finite point set. Comment: The answer is no. (E. D. Tymchatyn, 03/19/79) Problem 4. IfXAX is a Suslinean curve, does there exist a countable set in such thatAX intersects every nondegenerate subcontinuum of ? (A. Lelek, 09/22/71) Definition. A continuum isSuslinean if it does not contain uncountably many mutually exclusive nondegenerate subcontinua. Comment: No. (P. Minc, 06/15/79) 1 2 UNIVERSITY OF HOUSTON MATHEMATICS PROBLEM BOOK (10/14/94) Problem 5. Can every hereditarily decomposable chainable continuum be embedded in the plane in such a way that each end point is accessible from its complement? (H. Cook, 09/29/71) Definition. Asimple chain is a finite sequenceL,L,12··· ,Ln of open sets such thatLLij intersects if and only if|−|≤ ij 1. The terms of the sequence L,L,12··· ,Lnare called thelinks of the chain. An -chain is a chain each of whose links has diameter less thanM . A continuum ischainable if, for each positive numberM , can be covered by an -chain. Comment: No. (P. Minc and W. R. R. Transue, 02/27/90) Problem 6. Is it true that a chainable continuum can be embedded in the plane in such a way that every point is accessible from the complement if and only if it is Suslinean? (H. Cook, 09/29/71) Comment: That such a chainable continuum must be Suslinean follows from a result inBull. Polish Acad. Sci. 9 (1960), 271–276. (A. Lelek, 09/29/71) Comment: Yes. (P. Minc and W. R. R. Transue, 02/27/90) Problem 7. Is it true that the rim-type of any real (or half-ray) curve is at most 3? (A. Lelek, 09/29/71) Definition. A continuum is areal curve if it is a continuous 1-1 image of the real line and ahalf-ray curve if it is a continuous 1-1 image of the nonnegative reals. Definition. A continuum isrational if each two of its points can be separated by a countable point set. Definition. Suppose thatXa is a rational curve and is the least ordinal for which there exists a basisB forXa such that the -th derivative of the boundary of each element ofB is empty. ThenaX is therim-type of . Comment: A positive solution follows from Nadler’s work. (W. Kuperberg and P. Minc, 03/28/79) Problem 8. Does there exist a rational curve which topologically contains all real (or half-ray) curves? (A. Lelek, 09/29/71) Comment: Yes. (W. Kuperberg and P. Minc, 03/28/79) Problem 9. Does there exist a real curve which can be mapped onto each (weakly chainable) real curve? If not, does there exist a rational curve which can be mapped onto each (weakly chainable) real curve? (A. Lelek, 09/29/71) Definition. A continuum isweakly chainable if it is a continuous image of a chainable continuum. Comment: Delete the parentheses, weakly chainable is needed. (W. Kuperberg and P. Minc, 03/28/79) UNIVERSITY OF HOUSTON MATHEMATICS PROBLEM BOOK (10/14/94) 3 Problem 10. SupposeDS is a dendroid, is a compact, finitely generated commu- tative semigroup of monotone surjections onD which has a fixed end point. Does Shave another fixed point? (L. E. Ward, 10/20/71) Comment: Yes, even forλ -dendroids. (W. J. Gray and S. Williams, Bull. Polish Acad. Sci. 27 (1979), 599–604) Problem 11. Is each strongly Hurewicz space an absoluteFσ ? (A. Lelek, 11/03/71) Comment: No. (E. K. van Douwen, 05/06/79) Comment: The answer still seems to be unknown for the metric case. (A. Lelek, (08/31/94) Definition. A regular space is aHurewicz space if and only if, for each sequence of open coveringsY12 ,Y ,... , there exist finite subcollectionsAii of Y such that A12 ∪A ∪··· is a covering. A regular spaceX isstrongly Hurewicz if and only if, for each sequence of open coveringsY,Y,12··· of X , there exist finite subcollections ∗ AAiiofYX such that is the union over i of the intersection over j ofij+ . Notation. IfGG is a collection of sets, then∗ denotes the sum of all the sets in G . Problem 12. Is each product of two Hurewicz metric spaces a Hurewicz space? (A. Lelek, 11/03/71) Comment: No. (J. M. O’Farrell, 03/23/84) Problem 13. Is the product of two strongly Hurewicz metric spaces a strongly Hurewicz space? (H. Cook, 11/03/71) Problem 14. Is every Lindel¨of totally paracompact space Hurewicz? (D. W. Cur- tis, 11/10/71) Definition. A spaceXX istotally paracompact if every basis for has a locally finite subcollection coveringX . Problem 15. SupposefX is an open mapping of a compact metric space to the Sierpi´nski curve. Do there exist arbitrarily small closed neighborhoodsUxX of in for whichyfUfU is in Int() and| is confluent? (A. Lelek, 11/24/71) Definition. TheSierpi´nski curve is a locally connected planar curve with infin- itely many complementary domains whose boundaries are mutually exclusive simple closed curves. Definition. A mappingfX of a compact space onto a compact space Y is confluent if, for each continuumCY in , each component of fC−1 ( ) is mapped onto C by f . Comment: No. (T. Ma´ckowiak and E. D. Tymchatyn, 01/25/81) 4 UNIVERSITY OF HOUSTON MATHEMATICS PROBLEM BOOK (10/14/94) Problem 16. Supposef is a locally confluent and light mapping of a compact met- ric spaceX to the Sierpi´nski curve. Do there exist arbitrarily small closed neigh- borhoodsUxX of in for which y is in Int fUfU() and| is confluent? (A. Lelek, 11/24/71) Definition. A mappingfXY from a space onto a space islocally confluent if, for each pointyY of , there is an open set U containing y such that fU−1 ( ) is confluent. Definition. A mapping islight if each point inverse is totally disconnected. Comment: Yes. (T. Ma´ckowiak and E. D. Tymchatyn, 01/25/81) Problem 17. Is there a spaceX on which the identity is strongly homotopically stable but there is an open mapping ontoX which is not strongly homotopically stable? (H. Cook, 11/24/71) Definition. SupposefXY is a mapping of a metric space into a metric space and aXfis a point of . Then is (1)unstable ata>gX provided, for each 0, there exists a mapping from intoYdfx,gx<xXfagX such that ( ( ) ( )) for in and ( ) in ( ); (2)homotopically unstable ata> provided, for each 0, there exists a homotopyhX of× [0 , 1] into Y such that if x is in X and t is in I = [0 , 1], then h( x, 0) = f ( x ) , d h ( x, 0) , h ( x, t ) < , and f ( a )6 = h ( x, 1); (3)strongly homotopically stable atah provided for each homotopy of X× Iinto Y , where hx, ( 0) = fx ( ), we have ha,t ( ) = fa ( ) for t in I . Problem 18. Is it true that each open mapping of a continuumX onto the Sierp- i´nski curve (or the Menger curve) is strongly homotopically stable at each point of X? (A. Lelek, 11/24/71) Comment: No. (T. Ma´ckowiak and E. D. Tymchatyn, 01/25/81) Problem 19. Does each strictly non-mutually aposyndetic continuum with no weak cut point contain uncountably many mutually exclusive triods? (H. Cook, 12/08/71) Definition. A continuumMA ismutually aposyndetic if, for each two points andBM of , there exist mutually exclusive subcontinua H and KM of containing ABand , respectively, in their interiors. A continuum is strictly non-mutually aposyndetic if each two of its subcontinua with interiors intersect. Definition. A pointxM of a continuum is aweak cut point of M if there are two pointspqM and of such that every subcontinuum of M that contains both p andqx also contains . Comment: E. E. Grace has shown that every planar strictly non-mutually aposyn- detic continuum has a weak cut point. UNIVERSITY OF HOUSTON MATHEMATICS PROBLEM BOOK (10/14/94) 5 Problem 20. SupposefX is an open continuous mapping of a continuum onto YXXXX. Does there exist a space∗∗∗ such that is a subset of , is a locally connected continuum, and there is an extensionff∗∗∗ of from XY to such that ffXfXXX∗∗is open and() does not intersect ∗∗∗ (\ ) ? Can be the Hilbert cube? IfXX is a curve, can∗ be the Menger universal curve? (A. Lelek, 12/08/71) Comment: The answer to the first and the last question is yes.
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