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Open problems in topology
Topology and its Applications 38 (1991) 101-105 North-Holland
Open problems in topology J. van Mill (Editor) Free University,Amsterdam, Netherlands
G.M. Reed (Editor) St Edmund Hall, Oxford OX1 4AR, England
This is the first in a series of status reports on the 1100open problems listed in the book Open Problemsin Topology (North-Holland, Amsterdam, 1990), edited by the authors. For the next few years, Topologyand its Appkutions will publish such status repurts at regular intervals. Our goal is to give a brief account of problems solved, by whom, and whether preprints are available. Already more than twenty sohttions have been announced. In addition to announcements of new solutions, thesereportswill contain a matrix of problem numbers updated to indicate those problems still open. In this manner, we hope to fulfill our initial ambition to provide a source book of current open problems. Of course, this will only be possible, if we have the co-operation of both problem solvers and problem posers. To increase our chances of success, we would like to request each reader who solves a problem to notify both the author(s) of the paper in which it appeared in the book and one of the two editors of this section. As mentioned in the book, we plan a complete revision with the addition of new topics and authors within five years. We hope that this format will prove a useful service to the research community in topology. Problem 5
partially solved (consistently yes) by A. Dow (Canada); a preprint is available.
Problem 8
solved by S. Todorcevic (currently in Canada); there is no preprint yet.
Problem 13
solved in the negative by A. Dow (Canada) and EL Frankiewicz (Poland); a preprint is available.
Problem 16
solved by I. Juhisz and Z. Szentmiklossy (Hungary); a preprint is available.
Problem 23
solved by A. Krawczyk (Poland); preprints are available.
Problem 24 Problem 30
solved by A. Krawczyk (Poland); preprints are available. solved by S, Shelah (Israel) and J, Steprans (Canada); a preprint is available.
102
J. uun Mill, GM.
Reed
Open prob/llm.\" in topology
560 573 586 599 612 625 638 651 664 677 690
703 716 729 742 755 768 781 794 807 820 833 846 859 872 885 898 911 924 937 950 963 976 989 1002 1015 1028 1041 1054 1067 1080 1093
561 574 587 6000 613 626
562 575 588 601 614 627 ",,63:9;, 640 652 653 665 666 678 679 691 692 704 705 717 718 730 731 743 744 756 757 769 770 782 783 795 796 808 809 821 822 834 835 847 848 860 861 873 874 886 887 899 900 912 913 925 92,6 938 939 951 952 964 965 977 978 990 991 1003 1004 1016 1017 1029 1030 1042 1043 1055 1056 1068 1069 1081 1082 1094 1095
563 576 589 602 615 628 641 654 667
564 565 566 567 568 569 570 577 578 579 580 581 582 583 590 59'1 592 593 594 595 596, 603 604 605 606 607 608 609 616 617 618 619 620 621 622 629 630 631 632 633 634 635 642 643 ' :!~~~f.f 645 646 647 648 655 656 657 658 659 660 661 668 669 670 ' 671 672 673 674 680 681 682 683 684 685 686 687 693 694 695 696 697 698 699 700 Jr. ', 'j':1+~~1f! 706 707 708 709 "'-'I :.,;. R ,tt ~tfZ~i ~ J 713 ~~1$1" 720 721 722 723 724 725 726 732 733 734 735 736 737 738 739 745 746 747 748 749 750 751 752 758 759 760 761 762 763 764 765 771 772 773 774 775 776 777 778 784 785 786 787 788 789 790 791 797 798 799 BOO 801 802 803 804 810 811 812 813 814 815 816 817 823 824 825 826 827 828 829 830 836 837 838 839 840 841 842 843 849 850 851 852 853 . 854 855 856 , 862 863 864 865 866 867 868 869 875 876 877 878 879 880 881 882 888 889 890 891 892 893 894 895 901 902 903 904 905 906 907 908 914 915 916 917 918 919 920 921 927' 928 929 930 931 932 933 934 940 941 942 943 944 945 946 947 953 954 955 956 957 958 959 960 966 967 968 969 970 971 972 973 979 980 981 '982 983 984 985 986 992 993 994 995 996 997 998 999 1005 1006 1007 1008 1009 1010 1011 1012 1018 1019 1020 1021 1022 1023 1024 1025 1031 1032 1033 1034 1035 1036 1037 1038 1044 1045 1046 1047 1048 1049 1,050 1051 1057 1058 1059 1060 1061 ,1062 1063 1064 1070 .1071 1072 1073 1074 ' 1075 1076 1077 1083 1084 1085 1086 1087 1088 1089 1090 1096 1097 1098 1099 11'00 '" ....1' ,...1.
.. _11:
103
571 584 597 610 623 636 '$4, ~~ 662 675 688 701
.'1',
]1.4,~,
727 740 753 766 779
792 805 818 831 844 857 870 883 896 909 ' 922 935 948 961
974 987 1000 1013 1026 1039 1052 1065 1078 1091
572 585 598 61 1 624 637 650 663 676 689 702 715 728 741 754 767 780 793 806 819 832 845 858 871 884 897 910 923 936 949
962
975 988 1001 1014 1027 1040 1053 1066, 1079 1092
J. van Mill, G.M. Reed
104
Problem 179
solved by P. Koszmider (Canada); a preprint is available.
Problem 216
solvedfin the negative by A. Dow (Canada); a preprint is available.
Problem 237
solved by D. Strauss (UK) in the negative; a preprint is available.
Problem 299
G.M. Reed (UK) has constructed a normal Moore space whose square is not normal consistent with GCH. A preprint will be available soon.
Problem 305
In the second paragraph from the bottom on page 170, the cardinal “b” should be replaced by “b”. That is, (b = 0,) instead of (b = 0,); (b = c) instead of (b = c); and (b < w(X) CC) instead of (b < w(X) c C).
Problem 306
solved by S. Todorcevic (currently in Canada). The editors have a handwritten preprint.
Problem 3 14
solved by I. Tree (UK) and S. Watson (Canada) with-an example under CH. Also Tree has given a ZFC exampleyof a pseudo-normal Moore manifold which is not met&able (which is listed as a subproblem to 314). A preprint will be available shortly.
Problem 318
solved by G. Gruenhage (USA) and 2. Balogh (Hungary). On our question as to whether a preprint is available, M.E. Rudin answered, “consistently yes”.
Problem 339
there is a typo in the statement of the problem: the typo is “ts add(lJ6)” should be 9 c add@)“.
Problem 345
the claim (attributed to Nyikos) following Problem 345 has been withdrawn.
Problem 359
I. Tre ;-3K) has made an observation towards the answer of this problem. He noticed that the consistency of a, c e follows from two known results:
0) assuming “b = o,“, van Douwen (The integers and topology,
in: K. Kunen and I.E. Vaughan, editors, Handbook of SetTheoretic Topology ( NorthaHolland, Amsterdam) pages 11l_ 167) (see the proof of 11.4(e)) has constructed a first countable, pseudocompact not countably compact space, and (2) Shelah (On cardinal invariants of the continuum, in: J.E. Baumgartner, D.A. Martin and S. Shelah, editors, Axiomatic Set 7%eoe, Contemporary Mathematics 31 (American Mathematical Society, Providence, RI) pages 183~207) has constructed a -model where ddlf b C a.
Open problems in topology
Problem 397
105
In Remark 3.9 on page 276, the last sentence should read as follows: If the answer to Question 3.2 is “yes”, and if B is separable, then such an f exists even when X carries the weak* topologyand Y the norm topology. Note: This sentence differs from the one which appears now in the article by the addition of the phrase “and if B
is separable”. In the reference on page 278, “Raymond, J.S.” should read “Saint Raymond, J.“. Problem 407 Problem 424
solved by J. Dydak and J. Walsh (USA); a preprint is available. See also Problems 649, 7 12, and 7 14. solved by A.N. Dranishnikov. See his paper in Topology and its Applications35 (1990) pages 71-73.
Problem 460
partially solved by P. Krupski (Poland) and J. Rogers (USA); Yes, for finitely cyclic curves. A preprint is available.
Problem 639
was solved by R. Daverman (USA). The two classes referred to are different. There is no preprint available.
Problem 644
solved by L. Montejano (Mexico), as /was noted in a footnote in the ,book. A preprint is available.
Problem 649
solved by J. Dydak and J. Walsh (USA); a preprint is available. See..also Problems 712 and 714.
Problem 710 Problem 711
solved by T. Rushing and R. Sher (USA); a preprint is available. solved by L. Rubin (USA); a preprint is available.
Problem 7 12
solved by J. Dydak and J. Walsh (USA); preprints are available.
Problem 714
solved by J. Dydak and J. Walsh (USA); preprints are available.
Problem 719
solved by P. Mrozik (Germany); a preprint is avai#lable.
Problem 1070 should read: “Can a zero-dimensionalcompactpartial two-pointset always be extended to a two-pointset”. For noncompact partial two-point sets a counterexample was constructed by J. Kulesza (USA).
Open problems in topology J. van Mill (Editor) Free University,Amsterdam, Netherlands
G.M. Reed (Editor) St Edmund Hall, Oxford OX1 4AR, England
This is the first in a series of status reports on the 1100open problems listed in the book Open Problemsin Topology (North-Holland, Amsterdam, 1990), edited by the authors. For the next few years, Topologyand its Appkutions will publish such status repurts at regular intervals. Our goal is to give a brief account of problems solved, by whom, and whether preprints are available. Already more than twenty sohttions have been announced. In addition to announcements of new solutions, thesereportswill contain a matrix of problem numbers updated to indicate those problems still open. In this manner, we hope to fulfill our initial ambition to provide a source book of current open problems. Of course, this will only be possible, if we have the co-operation of both problem solvers and problem posers. To increase our chances of success, we would like to request each reader who solves a problem to notify both the author(s) of the paper in which it appeared in the book and one of the two editors of this section. As mentioned in the book, we plan a complete revision with the addition of new topics and authors within five years. We hope that this format will prove a useful service to the research community in topology. Problem 5
partially solved (consistently yes) by A. Dow (Canada); a preprint is available.
Problem 8
solved by S. Todorcevic (currently in Canada); there is no preprint yet.
Problem 13
solved in the negative by A. Dow (Canada) and EL Frankiewicz (Poland); a preprint is available.
Problem 16
solved by I. Juhisz and Z. Szentmiklossy (Hungary); a preprint is available.
Problem 23
solved by A. Krawczyk (Poland); preprints are available.
Problem 24 Problem 30
solved by A. Krawczyk (Poland); preprints are available. solved by S, Shelah (Israel) and J, Steprans (Canada); a preprint is available.
102
J. uun Mill, GM.
Reed
Open prob/llm.\" in topology
560 573 586 599 612 625 638 651 664 677 690
703 716 729 742 755 768 781 794 807 820 833 846 859 872 885 898 911 924 937 950 963 976 989 1002 1015 1028 1041 1054 1067 1080 1093
561 574 587 6000 613 626
562 575 588 601 614 627 ",,63:9;, 640 652 653 665 666 678 679 691 692 704 705 717 718 730 731 743 744 756 757 769 770 782 783 795 796 808 809 821 822 834 835 847 848 860 861 873 874 886 887 899 900 912 913 925 92,6 938 939 951 952 964 965 977 978 990 991 1003 1004 1016 1017 1029 1030 1042 1043 1055 1056 1068 1069 1081 1082 1094 1095
563 576 589 602 615 628 641 654 667
564 565 566 567 568 569 570 577 578 579 580 581 582 583 590 59'1 592 593 594 595 596, 603 604 605 606 607 608 609 616 617 618 619 620 621 622 629 630 631 632 633 634 635 642 643 ' :!~~~f.f 645 646 647 648 655 656 657 658 659 660 661 668 669 670 ' 671 672 673 674 680 681 682 683 684 685 686 687 693 694 695 696 697 698 699 700 Jr. ', 'j':1+~~1f! 706 707 708 709 "'-'I :.,;. R ,tt ~tfZ~i ~ J 713 ~~1$1" 720 721 722 723 724 725 726 732 733 734 735 736 737 738 739 745 746 747 748 749 750 751 752 758 759 760 761 762 763 764 765 771 772 773 774 775 776 777 778 784 785 786 787 788 789 790 791 797 798 799 BOO 801 802 803 804 810 811 812 813 814 815 816 817 823 824 825 826 827 828 829 830 836 837 838 839 840 841 842 843 849 850 851 852 853 . 854 855 856 , 862 863 864 865 866 867 868 869 875 876 877 878 879 880 881 882 888 889 890 891 892 893 894 895 901 902 903 904 905 906 907 908 914 915 916 917 918 919 920 921 927' 928 929 930 931 932 933 934 940 941 942 943 944 945 946 947 953 954 955 956 957 958 959 960 966 967 968 969 970 971 972 973 979 980 981 '982 983 984 985 986 992 993 994 995 996 997 998 999 1005 1006 1007 1008 1009 1010 1011 1012 1018 1019 1020 1021 1022 1023 1024 1025 1031 1032 1033 1034 1035 1036 1037 1038 1044 1045 1046 1047 1048 1049 1,050 1051 1057 1058 1059 1060 1061 ,1062 1063 1064 1070 .1071 1072 1073 1074 ' 1075 1076 1077 1083 1084 1085 1086 1087 1088 1089 1090 1096 1097 1098 1099 11'00 '" ....1' ,...1.
.. _11:
103
571 584 597 610 623 636 '$4, ~~ 662 675 688 701
.'1',
]1.4,~,
727 740 753 766 779
792 805 818 831 844 857 870 883 896 909 ' 922 935 948 961
974 987 1000 1013 1026 1039 1052 1065 1078 1091
572 585 598 61 1 624 637 650 663 676 689 702 715 728 741 754 767 780 793 806 819 832 845 858 871 884 897 910 923 936 949
962
975 988 1001 1014 1027 1040 1053 1066, 1079 1092
J. van Mill, G.M. Reed
104
Problem 179
solved by P. Koszmider (Canada); a preprint is available.
Problem 216
solvedfin the negative by A. Dow (Canada); a preprint is available.
Problem 237
solved by D. Strauss (UK) in the negative; a preprint is available.
Problem 299
G.M. Reed (UK) has constructed a normal Moore space whose square is not normal consistent with GCH. A preprint will be available soon.
Problem 305
In the second paragraph from the bottom on page 170, the cardinal “b” should be replaced by “b”. That is, (b = 0,) instead of (b = 0,); (b = c) instead of (b = c); and (b < w(X) CC) instead of (b < w(X) c C).
Problem 306
solved by S. Todorcevic (currently in Canada). The editors have a handwritten preprint.
Problem 3 14
solved by I. Tree (UK) and S. Watson (Canada) with-an example under CH. Also Tree has given a ZFC exampleyof a pseudo-normal Moore manifold which is not met&able (which is listed as a subproblem to 314). A preprint will be available shortly.
Problem 318
solved by G. Gruenhage (USA) and 2. Balogh (Hungary). On our question as to whether a preprint is available, M.E. Rudin answered, “consistently yes”.
Problem 339
there is a typo in the statement of the problem: the typo is “ts add(lJ6)” should be 9 c add@)“.
Problem 345
the claim (attributed to Nyikos) following Problem 345 has been withdrawn.
Problem 359
I. Tre ;-3K) has made an observation towards the answer of this problem. He noticed that the consistency of a, c e follows from two known results:
0) assuming “b = o,“, van Douwen (The integers and topology,
in: K. Kunen and I.E. Vaughan, editors, Handbook of SetTheoretic Topology ( NorthaHolland, Amsterdam) pages 11l_ 167) (see the proof of 11.4(e)) has constructed a first countable, pseudocompact not countably compact space, and (2) Shelah (On cardinal invariants of the continuum, in: J.E. Baumgartner, D.A. Martin and S. Shelah, editors, Axiomatic Set 7%eoe, Contemporary Mathematics 31 (American Mathematical Society, Providence, RI) pages 183~207) has constructed a -model where ddlf b C a.
Open problems in topology
Problem 397
105
In Remark 3.9 on page 276, the last sentence should read as follows: If the answer to Question 3.2 is “yes”, and if B is separable, then such an f exists even when X carries the weak* topologyand Y the norm topology. Note: This sentence differs from the one which appears now in the article by the addition of the phrase “and if B
is separable”. In the reference on page 278, “Raymond, J.S.” should read “Saint Raymond, J.“. Problem 407 Problem 424
solved by J. Dydak and J. Walsh (USA); a preprint is available. See also Problems 649, 7 12, and 7 14. solved by A.N. Dranishnikov. See his paper in Topology and its Applications35 (1990) pages 71-73.
Problem 460
partially solved by P. Krupski (Poland) and J. Rogers (USA); Yes, for finitely cyclic curves. A preprint is available.
Problem 639
was solved by R. Daverman (USA). The two classes referred to are different. There is no preprint available.
Problem 644
solved by L. Montejano (Mexico), as /was noted in a footnote in the ,book. A preprint is available.
Problem 649
solved by J. Dydak and J. Walsh (USA); a preprint is available. See..also Problems 712 and 714.
Problem 710 Problem 711
solved by T. Rushing and R. Sher (USA); a preprint is available. solved by L. Rubin (USA); a preprint is available.
Problem 7 12
solved by J. Dydak and J. Walsh (USA); preprints are available.
Problem 714
solved by J. Dydak and J. Walsh (USA); preprints are available.
Problem 719
solved by P. Mrozik (Germany); a preprint is avai#lable.
Problem 1070 should read: “Can a zero-dimensionalcompactpartial two-pointset always be extended to a two-pointset”. For noncompact partial two-point sets a counterexample was constructed by J. Kulesza (USA).