Celebratio Mathematica — Rudin

[1] M. E. Estill: Concerning abstract spaces. Ph.D. thesis, University of Texas at Austin, 1949. Advised by R. L. Moore. A condensed version was published in Duke Math. J. 17:4 (1950). MR 2937954 phdthesis

[2] M. E. Estill: “Concerning abstract spaces,” Duke Math. J. 17 : 4 (1950), pp. 317–327. A condensed version of the author’s PhD thesis (1949). MR 42686 Zbl 0039.39303 article

[3] M. E. Estill: “Separation in non-separable spaces,” Duke Math. J. 18 : 3 (1951), pp. 623–629. MR 42687 Zbl 0044.19503 article

[4] M. E. Estill: “A primitive dispersion set of the plane,” Duke Math. J. 19 : 2 (1952), pp. 323–328. MR 48796 Zbl 0047.16402 article

[5] M. E. Estill: “Concerning a problem of Souslin’s,” Duke Math. J. 19 : 4 (1952), pp. 629–639. MR 50878 Zbl 0048.28401 article

In the first of Fundamenta Mathematicae, Souslin [1] raised the question of the existence of a connected, linearly ordered space which is not separable and does not contain uncountably many mutually exclusive segments. Let a space have property \( X \) if and only if it is not separable and does not contain uncountably many mutually exclusive domains. It is easily shown [3] that if there exists a linearly ordered space having property \( X \), there also exists a connected linearly ordered space having property \( X \).

The first three parts of R. L. Moore’s Axiom 1 of [2] state that:

There exists a sequence \( G_1 \), \( G_2 \), \( G_3,\dots \) such that (1) for each \( n \), \( G_n \) is a collection of regions covering \( S \), (2) for each \( n, G_{n+1} \) is a subcollection of \( G_n \), (3) if \( R \) is any region whatsoever, \( X \) is a point of \( R \) and \( Y \) is a point of \( R \) either identical with \( X \) or not, then there exists a natural number \( m \) such that if \( g \) is any region belonging to the collection \( G_m \) and containing \( X \) then \( \overline{g} \) is a subset of \( (R-Y)+X \).

Call this Axiom \( 1_3 \).

It is easily shown [4; p. 628, Theorem 12] that no linear space having property \( X \) satisfies Axiom \( 1_3 \). It has been shown in [5] that there exists a locally connected space satisfying Axiom \( 1_3 \) and having property \( X \), but that there does not exist a locally connected space satisfying Axiom \( 1_3 \) and having property \( X \) such that each two points can be separated by a finite point set. The theorems of this paper will prove that a necessary and sufficient condition that there exist a linear space having property \( X \) is that there exist a locally connected space satisfying Axiom \( 1_3 \) and having property \( X \) such that each two points can be separated by either a countable or a separable point set.

[6] M. E. Rudin: “Countable paracompactness and Souslin’s problem,” Can. J. Math. 7 (February 1955), pp. 543–547. MR 73155 Zbl 0065.38002 article

[7] M. E. Rudin: “A separable normal nonparacompact space,” Proc. Am. Math. Soc. 7 : 5 (October 1956), pp. 940–941. MR 81631 Zbl 0072.17704 article

A topological space \( X \) is said to be paracompact [Dieudonné 1944] if for every open covering \( G \) of \( X \) there is a locally finite open covering \( G^{\prime} \) of \( X \) which is a refinement of \( G \). (\( G^{\prime} \) is locally finite if every point of \( X \) has a neighborhood which intersects only a finite number of members of \( G^{\prime} \).) It is known that every paracompact Hausdorff space is normal [Dieudonné 1944] and that every metrizable space is paracompact [Stone 1948]. Since every normal Hausdorff space with a countable base is metrizable, therefore, every normal Hausdorff space with a countable base is paracompact.

The purpose of this paper is to show that the existence of a countable base cannot be replaced by separability in this last statement.

[8] M. E. Rudin: “A topological characterization of sets of real numbers,” Pac. J. Math. 7 : 2 (1957), pp. 1185–1186. MR 94774 Zbl 0079.16703 article

[9] M. E. Rudin: “A subset of the countable ordinals,” Am. Math. Monthly 64 : 5 (May 1957), pp. 351. MR 85202 Zbl 0109.24401 article

[10] M. E. Rudin and V. L. Klee, Jr.: “A note on certain function spaces,” Arch. Math. (Basel) 7 : 6 (1957), pp. 469–470. MR 88714 Zbl 0077.16401 article

[11] M. E. Rudin: “A property of indecomposable connected sets,” Proc. Am. Math. Soc. 8 : 6 (December 1957), pp. 1152–1157. MR 91446 Zbl 0081.16903 article

Two nonempty subsets of a topological space are said to be separated if neither intersects the closure of the other. A set is connected if it is not the union of two separated sets. A connected set \( I \) is indecomposable if it is not the union of two connected sets, neither of which is dense in \( I \).

In [1951], Swingle raised the following question: does there exist, in the plane, an indecomposable connected set \( I \), such that the set \( I\cup\{p\} \) fails to be indecomposable for some limit point \( p \) of \( I \)?

The purpose of this paper is to prove that the answer is negative. It is interesting to note that the plane plays an essential role here as the embedding space: if the plane is replaced by Euclidean 3-space in Swingle’s question, the answer turns out to be affirmative. The construction of such an example is rather complicated, and is not included in this paper.

[12] M. E. Rudin: “A connected subset of the plane,” Fund. Math. 46 : 1 (1958), pp. 15–24. MR 98356 Zbl 0227.54029 article

A subset of a topological space is said to be connected if it is not the union of two non-empty disjoint sets, neither of which contains a limit point of the other. A connected set is degenerate if it consists of a single point. The object of this paper is the construction of a connected set which has, in a certain sense, few connected subsets:

If the continuum hypothesis is true, then there exists a non-degenerate connected subset \( M \) of the plane with the following property: if \( N \) is a non-degenerate connected subset of \( M \), then \( M-N \) is at most countable.

This disproves a conjecture made by Erdős [1944, p. 443]. It might be of interest to note that the result cannot be strengthened, since every non-degenerate connected set \( M \) contains a non-degenerate connected subset \( N \), such that \( M-N \) is infinite [Erdős 1944, p. 443].

We shall first construct a certain indecomposable continuum \( I \) in the plane, discuss some of its properties, and then construct \( M \) as a subset of \( I \), by means of a transfinite induction process; \( M \) will contain at most one point on any composant of \( I \).

[13] M. E. Rudin: “An unshellable triangulation of a tetrahedron,” Bull. Am. Math. Soc. 64 : 3 (May 1958), pp. 90–91. MR 97055 Zbl 0082.37602 article

A triangulation \( K \) of a tetrahedron \( T \) is shellable if the tetrahedra \( K_1,\dots, K_n \) of \( K \) can be so ordered that \[ K_i\cup K_{i+1}\cup\cdots\cup K_n \] is homeomorphic to \( T \) for \( i=1,\dots,n \). Sanderson [Proc. Amer. Math. Soc. vol. 8 (1957) p. 917] has shown that, if \( K \) is a Euclidean triangulation of a tetrahedron then there is a subdivision \( K^{\prime} \) of \( K \) which is shellable; and he raises the question of the existence of a Euclidean triangulation of a tetrahedron which is unshellable. Such a triangulation will be described here.

Let \( T \) be a tetrahedron each of whose edges has length 1.

We will describe a nontrivial Euclidean triangulation \( K \) of \( T \) such that, if \( R \) is any tetrahedron of \( K \), then the closure of \( (T-R) \) is not homeomorphic to \( T \).

[14] M. E. Rudin: “Arcwise connected sets in the plane,” Duke Math. J. 30 : 3 (1963), pp. 363–366. MR 151942 Zbl 0131.38004 article

[15] M. E. Rudin: “A technique for constructing examples,” Proc. Am. Math. Soc. 16 : 6 (December 1965), pp. 1320–1323. MR 188976 Zbl 0141.20403 article

The word space in this paper will refer to Hausdorff spaces. I have recently been asked the following questions.

(by the topology class of R. H. Bing). Is there a regular, sequentially compact space in which some nested sequence of continua intersect in a disconnected set?
(by E. Michael). Is there a normal, sequentially compact but not compact, space having a separable, metric, locally compact, dense subset?

Examples showing that the answer to both questions is yes, modulo the continuum hypothesis, are easily constructed using a technique I have often used before. The technique, described in §I, is perhaps more interesting than the particular examples which are given in §II. §III gives a variation of the technique and raises some questions.

[16] M. E. Rudin: “Interval topology in subsets of totally orderable spaces,” Trans. Am. Math. Soc. 118 (1965), pp. 376–389. MR 179751 Zbl 0134.40802 article

A topological space is said to be totally orderable if the points of the space can be totally ordered in such a way that the interval topology induced by this ordering coincides with the given topology. Unfortunately, the property of being a totally orderable space is not hereditary.

Suppose that \( R \) is a topological space and that \( T \) is a subspace of \( R \) and that \( R \) is totally orderable. Theorem II of this paper will answer question 1963.4 of Nieuw Archief Voor Wiskunde by giving necessary and sufficient conditions for \( T \) to be a totally orderable space. However, the most interesting case is where \( R \) is the real line; the solution in this case is quite simple to state and understand and is given in Theorem I.

[17] M. E. Rudin: “Types of ultrafilters,” pp. 147–151 in Topology seminar (Madison, WI, 1965). Edited by R. H. Bing and R. J. Bean. Annals of Mathematics Studies 60. Princeton University Press, 1966. MR 216451 Zbl 0158.20104 incollection

Ralph J. Bean	Related
R H Bing	Related	Volume

[18] M. E. Rudin: “A new proof that metric spaces are paracompact,” Proc. Am. Math. Soc. 20 : 2 (February 1969), pp. 603. MR 236876 Zbl 0175.49702 article

[19] M. E. Rudin: “Souslin’s conjecture,” Am. Math. Monthly 76 : 10 (December 1969), pp. 1113–1119. MR 270322 Zbl 0187.27302 article

[20] M. E. Rudin: “Composants and \( \beta N \),” pp. 117–119 in Proceedings of the Washington State University conference on general topology (Pullman, WA, 25–27 March 1970). Edited by R. A. Stoltenberg and A. M. Carstens. Washington State University (Pullman, WA), 1970. MR 266162 Zbl 0194.54703 incollection

Allan Matlock Weber Carstens	Related
Ronald A. Stoltenberg	Related

[21] M. E. Rudin: “The box topology,” pp. 191–199 in Proceedings of the University of Houston point set topology conference (Houston, TX, 22–24 March 1971). Edited by D. R. Traylor. University of Houston, 1971. MR 407794 Zbl 0271.54002 incollection

[22] M. E. Rudin: “Partial orders on the types in \( \beta N \),” Trans. Am. Math. Soc. 155 : 2 (April 1971), pp. 353–362. MR 273581 Zbl 0212.54901 article

[23] M. E. Rudin: “A normal space \( X \) for which \( X\times I \) is not normal,” Bull. Am. Math. Soc. 77 : 2 (March 1971), pp. 246. An expanded version of this was published in Fund. Math 73:2 (1971–1972). MR 270328 Zbl 0206.51601 article

[24] M. E. Rudin: “Book review: ‘Counterexamples in topology’ by L. A. Steen and J. A. Seebach, Jr.,” Am. Math. Monthly 78 : 7 (August–September 1971), pp. 803–804. MR 1536430 article

J. Arthur Seebach, Jr	Related
Lynn Arthur Steen	Related

[25] M. E. Rudin: “A normal space \( X \) for which \( X\times I \) is not normal,” Fund. Math. 73 : 2 (1971–1972), pp. 179–186. A brief initial version was published in Bull. Am. Math. Soc. 77:2 (1971). MR 293583 Zbl 0224.54019 article

[26] M. E. Rudin: “A normal hereditarily separable non-Lindelöf space,” Ill. J. Math. 16 : 4 (1972), pp. 621–626. MR 309062 Zbl 0241.54013 article

[27] M. E. Rudin: “Box products and extremal disconnectedness,” pp. 274–283 in Proceedings of the University of Oklahoma topology conference (Norman, OK, 23–25 March 1972). Edited by D. C. Kay. University of Oklahoma (Norman, OK), 1972. conference dedicated to Robert Lee Moore. MR 370537 Zbl 0248.54007 incollection

David C. Kay	Related
R. L. Moore	Related	Volume

[28] M. E. Rudin: “The box product of countably many compact metric spaces,” General Topology Appl. 2 : 4 (December 1972), pp. 293–298. MR 324619 Zbl 0243.54015 article

[29] M. E. Rudin: “The normality of a product with a compact factor,” Bull. Am. Math. Soc. 79 : 5 (September 1973), pp. 984–985. MR 331322 Zbl 0267.54005 article

Assume all spaces are \( T_1 \) and all maps continuous.

Suppose \( X \) is compact and \( Y \) normal. Even if \( X \) is the closed unit interval there is no reason to expect \( X\times Y \) to be normal [Rudin 1971]. Classically we have: \( X\times Y \) is normal if \( Y \) is paracompact [Dieudonné 1944]; \( \beta Y\times Y \) is normal only if \( Y \) is paracompact [Tamano 1960]; and \( [0,1]\times Y \) is normal if and only if \( Y \) is countably paracompact [Dowker 1951]. The purpose of this paper is to announce that the following conjecture of Morita [1961–1962], [Nagami 1971] is true.

If \( X \) is compact, \( X\times Z \) normal, and \( Y \) the image of \( Z \) under a closed map, then \( X\times Y \) is normal.

[30] M. E. Rudin: “Two problems,” pp. 221 in Proceedings of the international symposium on topology and its applications (Budva, Yugoslavia, 25–31 August 1972). Edited by D. R. Kurepa. Savez Društava Matematičara, Fizičara i Astronoma (Belgrade), 1973. incollection

[31] M. E. Rudin: “Countable box products of ordinals,” Trans. Am. Math. Soc. 192 (1974), pp. 121–128. MR 340022 Zbl 0289.02052 article

[32] M. E. Rudin: “A non-normal hereditarily-separable space,” Ill. J. Math. 18 : 3 (1974), pp. 481–483. MR 343235 Zbl 0282.54010 article

[33] M. E. Rudin: “Souslin trees and Dowker spaces,” pp. 557–562 in Topics in topology (Keszthely, Hungary, 19–23 June 1972). Edited by Á. Czászár. Colloquia Mathematica Societatis János Bolyai 8. North-Holland (Amsterdam), 1974. MR 365483 Zbl 0293.54041 incollection

[34] M. E. Rudin and M. Starbird: “Products with a metric factor,” General Topology Appl. 5 : 3 (September 1975), pp. 235–248. MR 380709 Zbl 0305.54010 article

[35] P. Erdős and M. E. Rudin: “A non-normal box product,” pp. 629–631 in Infinite and finite sets: Dedicated to P. Erdős on his 60th birthday (Keszthely, Hungary, 1973), vol. 2. Edited by A. Hajnal, R. Rado, and V. T. Sos. Colloquia Mathematica Societatis János Bolyai 10. North-Holland (Amsterdam), 1975. MR 394537 Zbl 0328.54017 incollection

Richard Rado	Related
Paul Erdős	Related
Vera T. Sós	Related
András Hajnal	Related

[36] M. E. Rudin: Lectures on set theoretic topology (Laramie, WY, 12–16 August 1974). CBMS Regional Conference Series in Mathematics 23. American Mathematical Society (Providence, RI), 1975. Reprinted in 1980. MR 367886 Zbl 0318.54001 book

[37] M. E. Rudin: “The normality of products,” pp. 81–84 in Proceedings of the International Congress of Mathematicians 1974 (Vancouver, BC, 21–29 August 1974), vol. 2. Edited by R. D. James. Canadian Mathematical Congress (Montreal), 1975. MR 425879 Zbl 0339.54016 incollection

[38] M. E. Rudin: “The normality of products with one compact factor,” General Topology Appl. 5 : 1 (April 1975), pp. 45–59. MR 362214 Zbl 0296.54004 article

[39] M. E. Rudin: “A separable Dowker space,” pp. 125–132 in Topologia insiemistica e generale, gruppi topologici e gruppi di Lie [Set theoretic and general topology, topological groups and Lie groups] (Rome, 8–13 March 1973 and 22–25 January 1974). Symposia Mathematica 16. Academic Press (New York), 1975. MR 415584 Zbl 0317.54046 incollection

[40] M. E. Rudin: “The metrizability of normal Moore spaces,” pp. 507–516 in Studies in topology (Charlotte, NC, 14–16 March 1974). Edited by N. M. Stavrakas and K. R. Allen. Academic Press (New York), 1975. MR 358711 Zbl 0307.54028 incollection

Keith Roy Allen	Related
Nicholas Mark Stavrakas	Related

[41] M. E. Rudin and P. Zenor: “A perfectly normal nonmetrizable manifold,” Houston J. Math. 2 : 1 (1976), pp. 129–134. MR 394560 Zbl 0315.54028 article

[42] I. Juhász, K. Kunen, and M. E. Rudin: “Two more hereditarily separable non-Lindelöf spaces,” Can. J. Math. 28 : 5 (October 1976), pp. 998–1005. MR 428245 Zbl 0336.54040 article

Kenneth Kunen	Related
István Juhász	Related

[43] M. E. Rudin and M. Starbird: “Some examples of normal Moore spaces,” Can. J. Math. 29 : 1 (February 1977), pp. 84–92. MR 448311 Zbl 0341.54038 article

[44] Y. Benyamini, M. E. Rudin, and M. Wage: “Continuous images of weakly compact subsets of Banach spaces,” Pac. J. Math. 70 : 2 (1977), pp. 309–324. MR 625889 Zbl 0374.46011 article

Yoav Benyamini	Related
Michael Wage	Related

[45]J. M. Aarts and E. Lowen-Colebunders: “On an example of Mary Ellen (Estill) Rudin,” pp. 18–20 in General topology and its relations to modern analysis and algebra, IV (Prague, 1976), part B. Soc. Czechoslovak Mathematicians and Physicists (Prague), 1977. MR 0461452 Zbl 0383.54020

Eva Lowen-Colebunders	Related
Johannes Michael Aarts	Related

[46] M. E. Rudin: “A narrow view of set theoretic topology,” pp. 190–195 in General topology and its relations to modern analysis and algebra, IV (Prague, August 1976), part A: Invited papers. Edited by J. Novák. Lecture Notes in Mathematics 609. Springer (Berlin), 1977. Proceedings of the fourth Prague topological symposium, 1976. MR 493931 Zbl 0367.54001 incollection

[47] M. E. Rudin: “Martin’s axiom,” pp. 491–501 in Handbook of mathematical logic, part B: Set theory. Edited by J. Barwise. Studies in Logic and Foundations of Mathematics 90. North-Holland (Amsterdam), 1977. MR 540758 incollection

[48] E. A. Michael and M. E. Rudin: “A note on Eberlein compacts,” Pac. J. Math. 72 : 2 (1977), pp. 487–495. MR 478092 Zbl 0345.54020 article

[49] E. Michael and M. E. Rudin: “Another note on Eberlein compacts,” Pac. J. Math. 72 : 2 (1977), pp. 497–499. MR 478093 Zbl 0344.54018 article

[50] M. E. Rudin: “\( \aleph \)-Dowker spaces,” Czechoslovak Math. J. 28 : 2 (1978), pp. 324–326. MR 478098 Zbl 0383.54012 article

In a written communication to the Prague Topology Symposium of 1976, K.Morita proposed the following:

Conjecture 1. If a Hausdorff space \( Y \) has the property that \( X\times Y \) is normal for all normal Hausdorff spaces \( X \), then \( Y \) is discrete.

In an abstract and talk at this symposium M. Atsuji pointed out that Morita’s conjecture follows from:

Conjecture 2. For each infinite cardinal \( \aleph \), there is a normal Hausdorff space \( X_{\aleph} \) which has a decreasing family \[ \{D_{\alpha}\}_{\alpha < \aleph} \] of closed sets such that \[ \bigcap_{\alpha < \aleph}D_{\alpha} = \emptyset \] and, if \[ \{U_{\alpha}\}_{\alpha < \aleph} \] is a family of open sets with \( D_{\alpha}\subset U_{\alpha} \) for each \( \alpha \), then \[ \bigcap_{\alpha < \aleph}U_{\alpha}\neq \emptyset .\]

A space \( X_{\aleph} \) having the properties described in Atsuji’s conjecture could be called a \( \aleph \)-Dowker space since \( X_{\omega} \) would be an ordinary Dowker space. The purpose of this note is to prove that there are \( \aleph \)-Dowker spaces for all cardinals \( \aleph \), thus proving conjectures (1) and (2).

[51] S. Shelah and M. E. Rudin: “Unordered types of ultrafilters,” Topology Proc. 3 : 1 (1978), pp. 199–204. MR 540490 Zbl 0431.03033 article

[52] M. E. Rudin: “Book reviews: C. O. Christenson and W. L. Voxman,‘Aspects of topology’ and G. L. Naber, ‘Sset-theoretic topology: With emphasis on problems from the theory of coverings, zero dimensionality, and cardinal invariants’,” Bull. Am. Math. Soc. 84 : 2 (March 1978), pp. 271–272. MR 1567046 article

William Voxman	Related
Gregory L. Naber	Related
Charles O. Christenson	Related

[53] M. E. Rudin: “The undecidability of the existence of a perfectly normal nonmetrizable manifold,” Houston J. Math. 5 : 2 (1979), pp. 249–252. MR 546759 Zbl 0418.03036 article

[54] M. E. Rudin: “Hereditary normality and Souslin lines,” General Topology Appl. 10 : 1 (February 1979), pp. 103–105. MR 519717 Zbl 0405.54017 article

[55] M. E. Rudin: “Pixley–Roy and the Souslin line,” Proc. Am. Math. Soc. 74 : 1 (1979), pp. 128–134. MR 521886 Zbl 0414.54003 article

[56] L. Smail: Interview with Mary Ellen Rudin, 1979. University of Wisconsin at Madison, online MP3 audio, three parts. misc

[57] M. E. Rudin: “\( S \) and \( L \) spaces,” pp. 431–444 in Surveys in general topology. Edited by G. M. Reed. Academic Press (New York), 1980. MR 564109 Zbl 0457.54015 incollection

[58] M. E. Rudin: Lectures on set theoretic topology (Laramie, WY, 12–16 August 1974), Reprinted edition. CBMS Regional Conference Series in Mathematics 23. American Mathematical Society (Providence, RI), 1980. Reprint of 1975 original. Zbl 0472.54001 book

[59] M. E. Rudin: “Directed sets which converge,” pp. 305–307 in General topology and modern analysis (Riverside, CA, 28–31 May 1980). Edited by L. F. McAuley and M. M. Rao. Academic Press (New York), 1981. MR 619055 Zbl 0457.04007 incollection

Louis F. McAuley	Related
Malempati Madhusudana Rao	Related

[60] M. E. Rudin and S. Watson: “Countable products of scattered paracompact spaces,” Proc. Am. Math. Soc. 89 : 3 (1983), pp. 551–552. MR 715885 Zbl 0518.54021 article

[61] M. E. Rudin: “The shrinking property,” Can. Math. Bull. 26 : 4 (1983), pp. 385–388. MR 716576 Zbl 0536.54013 article

[62] M. E. Rudin: “Collectionwise normality in screenable spaces,” Proc. Am. Math. Soc. 87 : 2 (1983), pp. 347–350. MR 681846 Zbl 0502.54018 article

[63] M. E. Rudin: “Dowker’s set theory question,” Quest. Answers Gen. Topology 1 : 2 (1983), pp. 75–76. MR 722085 Zbl 0536.54003 article

[64] M. E. Rudin: “Yasui’s questions,” Quest. Answers Gen. Topology 1 : 2 (1983), pp. 122–127. MR 722092 article

[65] M. E. Rudin: “A normal screenable non-paracompact space,” Topology Appl. 15 : 3 (May 1983), pp. 313–322. MR 694550 Zbl 0516.54004 article

[66] M. E. Rudin: “Dowker spaces,” pp. 761–780 in Handbook of set-theoretic topology. Edited by K. Kunen and J. E. Vaughan. North-Holland (Amsterdam), 1984. MR 776636 Zbl 0554.54005 incollection

Kenneth Kunen	Related
Jerry Eugene Vaughan	Related

[67] M. E. Rudin: “Two problems of Dowker,” Proc. Am. Math. Soc. 91 : 1 (1984), pp. 155–158. MR 735583 Zbl 0554.54006 article

[68] A. Jackson: “Mary Ellen Rudin” in Profiles of women in mathematics: The Emmy Noether Lectures. Association for Women in Mathematics (College Park, MD), 1984. incollection

[69] M. E. Rudin: “\( \kappa \)-Dowker spaces,” pp. 175–193 in Aspects of topology: In memory of Hugh Dowker 1912–1982. Edited by I. M. James and E. H. Kronheimer. London Mathematical Society Lecture Note Series 93. Cambridge University Press, 1985. MR 787828 Zbl 0566.54009 incollection

Ioan M. James	Related
Erwin H. Kronheimer	Related
Clifford Hugh Dowker	Related

[70] A. Bešlagić and M. E. Rudin: “Set-theoretic constructions of nonshrinking open covers,” Topology Appl. 20 : 2 (August 1985), pp. 167–177. MR 800847 Zbl 0574.54020 article

A family \[ \{M_{\alpha}\mid \alpha\in A\} \] is a shrinking of a cover \[ \{O_{\alpha}\mid \alpha\in A\} \] of a topological space if \[ \{M_{\alpha}\mid \alpha\in A\} \] also covers and \( M_{\alpha}\subset O_{\alpha} \) for all \( \alpha\in A \).

\( \lozenge^{++} \) implies that there is a normal space such that every increasing open cover of it has a clopen shrinking but there is an open cover having no closed shrinking.

\( \lozenge \) implies that there is a \( P \)-space (i.e. a space having a normal product with every metric space), which has an increasing open cover having no closed shrinking. This space is used in [Chiba et al. 1986] to show that any space which has a normal product with every \( P \)-space is metrizable.

[71] P. J. Collins, G. M. Reed, A. W. Roscoe, and M. E. Rudin: “A lattice of conditions on topological spaces,” Proc. Am. Math. Soc. 94 : 3 (1985), pp. 487–496. MR 787900 Zbl 0562.54043 article

A. William Roscoe	Related
George Michael Reed	Related
Peter J. Collins	Related

[72] K. Chiba, T. C. Przymusiński, and M. E. Rudin: “Normality of product spaces and Morita’s conjectures,” Topology Appl. 22 : 1 (February 1986), pp. 19–32. MR 831178 Zbl 0575.54007 article

It is well-known that \( Z \) is a perfectly normal space (normal \( P \)-space) if and only if \( X\times Z \) is perfectly normal (normal) for every metric space \( X \). Conversely, denote by \( \mathbf{Q} \) (resp. \( \mathbf{N} \)) the class of all spaces \( X \) whose products \( X\times Z \) with all perfectly normal spaces (all normal \( P \)-spaces) \( Z \) are normal. It is natural to ask whether \( \mathbf{Q} \) and \( \mathbf{N} \) necessarily coincide with the class \( \mathbf{M} \) of metrizable spaces.

Clearly, \( \mathbf{M}\subset\mathbf{N}\subset\mathbf{Q} \). We prove that first countable members of \( \mathbf{Q} \) are metrizable and that under \( V=L \) the classes \( \mathbf{M} \) and \( \mathbf{N} \) coincide, thus giving a consistency proof of Morita’s conjecture. On the other hand, even though \( \mathbf{Q} \) contains non-metrizable members, it is quite close to \( \mathbf{M} \): the class \( \mathbf{Q} \) is countably productive and hereditary, and all members \( X \) of \( \mathbf{Q} \) are stratifiable and satisfy \[ c(X)=l(X)=w(X) .\] In particular, locally Lindelöf or locally Souslin or locally \( p \)-spaces in \( \mathbf{Q} \) are metrizable.

The above results immediately lead to the consistency proof of another Morita’s conjecture, stating that \( X \) is a metrizable \( \sigma \)-locally compact space if and only if \( X\times Y \) is normal for every normal countably paracompact space \( Y \). No additional set-theoretic assumptions are necessary if \( X \) is first countable.

In our investigation, an important role is played by the famous Bing examples of normal, non-collectionwise normal spaces. Answering Dennis Burke’s question, we prove that products of two Bing-type examples are always non-normal.

T. C. Przymusiński	Related
K. Chiba	Related

[73] R. McCroskey Karr, J. Rezaie, and J. E. Wilson: “Mary Ellen Rudin (1924–),” pp. 190–192 in Women of mathematics: A bibliographical sourcebook. Edited by L. S. Grinstein and P. J. Campbell. Greenwood Press (Westport, CT), 1987. MR 911515 incollection

Joel E. Wilson	Related
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[74] D. J. Albers and C. Reid: “An interview with Mary Ellen Rudin,” College Math. J. 19 : 2 (March 1988), pp. 114–137. MR 944786 Zbl 0995.01519 article

Donald J. Albers	Related
Constance Reid	Related

[75] M. E. Rudin: “A few topological problems,” Commentat. Math. Univ. Carol. 29 : 4 (1988), pp. 743–746. Dedicated to Professor M. Katetov on his seventieth birthday. MR 982794 Zbl 0688.54003 article

[76] M. E. Rudin: “A nonmetrizable manifold from \( \lozenge^+ \),” pp. 105–112 in Special issue on set-theoretic topology (York, ON, 9 February 1985), published as Topology Appl. 28 : 2. Issue edited by F. D. Tall. Elsevier (Amsterdam), March 1988. MR 932975 Zbl 0637.54004 incollection

[77] M. E. Rudin: “Countable point separating open covers for manifolds,” Houston J. Math. 15 : 2 (1989), pp. 255–266. MR 1022067 Zbl 0719.54040 article

[78] M. E. Rudin: “Two nonmetrizable manifolds,” Topology Appl. 35 : 2–3 (June 1990), pp. 137–152. MR 1058794 Zbl 0708.54012 article

[79] M. E. Rudin: “Some conjectures,” Chapter 10, pp. 184–193 in Open problems in topology. Edited by J. van Mill and G. M. Reed. North-Holland (Amsterdam), 1990. MR 1078646 incollection

Jan van Mill	Related
George Michael Reed	Related

[80] “Mary Ellen Rudin,” pp. 282–302 in More mathematical people :Contemporary conversations. Edited by D. J. Albers, G. L. Alexanderson, and C. Reid. Academic Press (San Diego, CA), 1990. incollection

Donald J. Albers	Related
Constance Reid	Related
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[81] Z. Balogh, S. W. Davis, A. Dow, G. Gruenhage, P. J. Nyikos, M. E. Rudin, F. D. Tall, and S. Watson: “New classic problems,” Topol. Proc. 15 (1990), pp. 201–220. San Marcos, TX, 5–7 April 1990. Zbl 0779.54001 article

Mary Ellen Rudin and Frank Tall organized a problem session at the Spring Topology Conference in San Marcos, Texas in 1990 and invited several people to come up with their ideas for problems that should be the worthy successors to the S & L problems, the box product problems, the normal Moore space problem, etc. in the sense that they could and should be the focus of common activity during the 1990’s as the older problems had been during the 1970’s. They hoped that these problems would counterbalance the more centrifugal 1980’s, during which there was a tendency for each set-theoretic topologist to do his own thing, rather than there being many people working on problems generally recognized as important. This compilation is the result. Tinle will tell whether the title is appropriate.

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[82] Z. Balogh and M. E. Rudin: “Monotone normality,” Topology Appl. 47 : 2 (November 1992), pp. 115–127. MR 1193194 Zbl 0769.54022 article

[83] M. Starbird: “Mary Ellen Rudin as advisor and geometer,” pp. 114–118 in The work of Mary Ellen Rudin (Madison, WI, 26–29 June 1991). Edited by F. D. Tall. Annals of the New York Academy of Sciences 705. New York Academy of Sciences, 1993. Papers from the summer conference on general topology and applications. MR 1277884 Zbl 0814.01016 incollection

Michael Peter Starbird	Related
Franklin D. Tall	Related

[84] F. B. Jones: “Some glimpses of the early years,” pp. xi–xii in The work of Mary Ellen Rudin (Madison, WI, 26–29 June 1991). Edited by F. D. Tall. Annals of the New York Academy of Sciences 705. New York Academy of Sciences, 1993. Papers from the summer conference on general topology and applications. MR 1277877 incollection

Floyd Burton Jones	Related
Franklin D. Tall	Related

[85] N. Frank, F. D. Tall, R. Kopperman, P. Nyikos, and M. E. Rudin: “Reminiscences of Boris Shapirovskiĭ,” pp. xiii–xxi in Papers on general topology and applications: Seventh summer conference at the University of Wisconsin (Madison, WI, 26–29 June 1991). Edited by S. Andima, R. Kopperman, P. R. Misra, M. E. Rudin, and A. R. Todd. Annals of the New York Academy of Sciences 704. New York Academy of Sciences, 1993. MR 1277836 Zbl 0807.01031 incollection

Susan Jane Zimmerman Andima	Related
Boris Emilievich Shapirovskiĭ	Related
Aaron R. Todd	Related
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[86] Papers on general topology and applications: Seventh summer conference at the University of Wisconsin (Madison, WI, 26–29 June 1991). Edited by S. Andima, R. Kopperman, P. R. Misra, M. E. Rudin, and A. R. Todd. Annals of the New York Academy of Sciences 704. New York Academy of Sciences, 1993. MR 1277835 Zbl 0801.00034 book

Susan Jane Zimmerman Andima	Related
Aaron R. Todd	Related
Ralph David Kopperman	Related
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[87] The work of Mary Ellen Rudin (Madison, WI, 26–29 June 1991). Edited by F. D. Tall. Annals of the New York Academy of Sciences 705. New York Academy of Sciences, 1993. Papers from the summer conference on general topology and applications. MR 1277876 Zbl 0801.00017 book

[88] M. E. Rudin: “A cyclic monotonically normal space which is not \( K_0 \),” Proc. Am. Math. Soc. 119 : 1 (September 1993), pp. 303–307. MR 1185281 Zbl 0789.54027 article

[89] F. D. Tall: “The work of Mary Ellen Rudin,” pp. 1–16 in The work of Mary Ellen Rudin (Madison, WI, 26–29 June 1991). Edited by F. D. Tall. Annals of the New York Academy of Sciences 705. New York Academy of Sciences, 1993. Papers from the summer conference on general topology and applications. MR 1277878 Zbl 0837.54002 incollection

[90] P. J. Nyikos: “Mary Ellen Rudin’s contributions to the theory of nonmetrizable manifolds,” pp. 92–113 in The work of Mary Ellen Rudin (Madison, WI, 26–29 June 1991). Edited by F. D. Tall. Annals of the New York Academy of Sciences 705. New York Academy of Sciences, 1993. Papers from the summer conference on general topology and applications. MR 1277883 Zbl 0894.54003 incollection

Peter Joseph Nyikos	Related
Franklin D. Tall	Related

[91] S. Watson: “Mary Ellen Rudin’s early work on Suslin spaces,” pp. 168–182 in The work of Mary Ellen Rudin (Madison, WI, 26–29 June 1991). Edited by F. D. Tall. Annals of the New York Academy of Sciences 705. New York Academy of Sciences, 1993. Papers from the summer conference on general topology and applications. MR 1277887 Zbl 0849.54025 incollection

Steve Watson	Related
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[92] J. Roitman: “Conference in honor of Mary Ellen Rudin,” pp. ix–x in The work of Mary Ellen Rudin (Madison, WI, 26–29 June 1991). Edited by F. D. Tall. Annals of the New York Academy of Sciences 705. New York Academy of Sciences, 1993. Papers from the summer conference on general topology and applications. incollection

Judith Roitman	Related
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[93] M. E. Rudin and W. Rudin: “Continuous functions that are locally constant on dense sets,” J. Funct. Anal. 133 : 1 (October 1995), pp. 129–137. MR 1351645 Zbl 0916.46019 article

[94] M. E. Rudin: “A biconnected set in the plane,” Topology Appl. 66 : 1 (1995), pp. 41–48. MR 1357873 Zbl 0831.54003 article

[95] M. E. Rudin: “Monotone normality and compactness,” pp. 199–205 in Proceedings of the international conference on set-theoretic topology and its applications (Matsuyama, Japan, 12–16 December 1994), published as Topology Appl. 74 : 1–3. Issue edited by T. Nogura. Elsevier (Amsterdam), December 1996. MR 1425938 Zbl 0874.54004 incollection

[96] M. E. Rudin: “The early work of F. B. Jones,” pp. 85–96 in Handbook of the history of general topology, vol. 1. Edited by C. E. Aull and R. Lowen. Kluwer Academic (Dordrecht), 1997. Based on a lecture given at UC-Riverside in May 1992 and at an AMS meeting in San Antonio, January 1993. MR 1617573 Zbl 0954.54001 incollection

Floyd Burton Jones	Related
Robert Lowen	Related
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[97] S. Purisch and M. E. Rudin: “Products with linear and countable type factors,” Proc. Am. Math. Soc. 125 : 6 (1997), pp. 1823–1830. MR 1415363 Zbl 0917.54012 article

[98] M. E. Rudin, I. S. Stares, and J. E. Vaughan: “From countable compactness to absolute countable compactness,” Proc. Am. Math. Soc. 125 : 3 (March 1997), pp. 927–934. MR 1415367 Zbl 0984.54027 article

Ian S. Stares	Related
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[99] M. E. Rudin: “Zero dimensionality and monotone normality.” Edited by M. Hušek. Topology Appl. 85 : 1–3 (1998), pp. 319–333. MR 1617471 Zbl 0983.54007 article

[100] M. E. Rudin: “Compact, separable, linearly ordered spaces,” pp. 397–419 in Special volume in memory of Kiiti Morita, published as Topology Appl. 82 : 1–3. Issue edited by J.-i. Nagata, A. Okuyama, and T. Hoshina. Elsevier (Amsterdam), January 1998. MR 1602448 Zbl 0889.54014 incollection

Takao Hoshina	Related
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[101] M. E. Rudin: “Nikiel’s conjecture,” Topology Appl. 116 : 3 (December 2001), pp. 305–331. MR 1857669 Zbl 0988.54022 article

[102] M. E. Rudin: “Hereditarily paracompact and compact monotonically normal spaces,” pp. 179–189 in International school of mathematics “G. Stampacchia”: Convergence and topology (Erice, Sicily, 27 June–2 July 1998), published as Topology Appl. 111 : 1–2. Issue edited by P. Collins, S. Dolecki, and G. Tironi. 2001. MR 1806038 Zbl 0983.54021 incollection

Gino Tironi	Related
Peter J. Collins	Related
Szymon Dolecki	Related

[103] M. E. Rudin: “Topology in the 20th century,” pp. 565–568 in Recent progress in general topology, II (Prague, 19–25 August 2001). Edited by M. Hušek and J. van Mill. North-Holland (Amsterdam), 2002. MR 1970207 Zbl 1023.54001 incollection

Jan van Mill	Related
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[104] H. Bennett, D. Lutzer, and M. E. Rudin: “Lines, trees, and branch spaces,” Order 19 : 4 (2002), pp. 367–384. MR 1964446 Zbl 1025.54016 article

David John Lutzer	Related
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[105] M. E. Rudin: “A mathematical friendship.” Edited by D. Repovs and P. V. Semenov. Topology Appl. 155 : 8 (April 2008), pp. 752–753. MR 2406379 Zbl 1137.01325 article

Ernest A. Michael	Related
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P. V. Semenov	Related

[106] M. E. Rudin: “Peter and Mike.” Edited by C. Good, R. Knight, and B. Raines. Topology Appl. 156 : 11 (June 2009), pp. 1885. MR 2536168 Zbl 1165.01323 article

Peter J. Collins	Related
George Michael Reed	Related
Chris Good	Related
Robin William Knight	Related
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[107] M. E. Rudin: “Ken Kunen, a mathematician’s mathematician.” Edited by J. Hart. Topology Appl. 158 : 18 (December 2011), pp. 2444. MR 2847312 article

Kenneth Kunen	Related
Joan E. Hart	Related

[108] A. W. Miller: On the research of Mary Ellen Rudin, 2013. Two page document, online only. misc

[109] Special issue honoring the memory of Mary Ellen Rudin, published as Topology Appl. 195. Issue edited by G. Gruenhage and P. Nyikos. Elsevier (Amsterdam), November 2015. book

Gary Gruenhage	Related
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[110] G. Benkart, M. Džamonja, J. Roitman, I. Juhász, W. Fleissner, F. Tall, P. Nyikos, K. Kunen, and A. Miller: “Memories of Mary Ellen Rudin,” Notices Am. Math. Soc. 62 : 6 (2015), pp. 617–629. Coordinating editors were Georgia Benkart, Mirna Džamonja and Judith Roitman. MR 3362445 Zbl 1338.01028 article

István Juhász	Related
Kenneth Kunen	Related
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Mirna Džamonja	Related
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Franklin D. Tall	Related

[111] G. Gruenhage, P. Nyikos, I. Juhász, B. Fleissner, J. Hart, A. Ostaszewski, S. Mardešić, J. van Mill, J. Vaughan, F. Tall, and K. Yamazaki: “Mary Ellen Rudin — remembrances,” pp. 3–14 in Special issue honoring the memory of Mary Ellen Rudin, published as Topology Appl. 195. Issue edited by G. Gruenhage and P. Nyikos. Elsevier (Amsterdam), November 2015. Gary Gruenhage and Peter Nyikos were coordinating editors. MR 3414870 Zbl 1326.54005 incollection

Kaori Yamazaki	Related
Jerry Eugene Vaughan	Related
Gary Gruenhage	Related
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[112] G. Gruenhage and P. Nyikos: “Mary Ellen’s conjectures,” pp. 15–25 in Special issue honoring the memory of Mary Ellen Rudin, published as Topology Appl. 195. Issue edited by G. Gruenhage and P. Nyikos. Elsevier (Amsterdam), November 2015. MR 3414871 incollection

Gary Gruenhage	Related
Peter Joseph Nyikos	Related

[113] H. Bennett and D. Lutzer: “Mary Ellen Rudin and monotone normality,” pp. 50–62 in Special issue honoring the memory of Mary Ellen Rudin, published as Topology Appl. 195. Issue edited by G. Gruenhage and P. Nyikos. Elsevier (Amsterdam), November 2015. MR 3414874 Zbl 1327.54003 incollection

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Bibliography