Topology Atlas Document # taic-36 Topology Atlas Invited Contributions vol. 5 issue 1 (2000) 11-14

Recent Progress in the Topology of Generalized Ordered Spaces

Harold R Bennett and David J. Lutzer

Invited Contribution

The last decade has seen substantial advances in understanding the topology of linearly ordered and generalized ordered spaces.1 The most spectacular advance is due to Mary Ellen Rudin [Ru1] [Ru2] [Ru3] who solved Nikiel's problem affirmatively by proving:

1. Theorem: Every compact monotonically normal space is the continuous image of a compact LOTS.

A second major advance is due to Qiao and Tall [QT] who showed that several classical problems of Maurice, Heath, and Nyikos concerning perfect GO-spaces are equivalent by proving the next two theorems:

2. Theorem: The following statements are equivalent:

  1. there is a perfect LOTS that does not have a \sigma-closed-discrete dense subset;
  2. there is a perfectly normal, non-Archimedean space that is not metrizable;
  3. there is a LOTS X in which every disjoint collection of convex open sets is \sigma-discrete, and yet X does not have a \sigma-closed-discrete dense subset;
  4. there is a dense-in-itself LOTS Y that does not have a \sigma-closed-discrete dense set, and yet each nowhere dense subspace of Y does have a \sigma-closed-discrete dense subset (in its relative topology).

3. Theorem: The following statements are equivalent:

  1. there is a perfectly normal, non-metrizable, non-Archimedean space having a point-countable base;
  2. there is a perfect LOTS that has a point-countable base but does not have a \sigma-closed-discrete dense subset;
  3. there is a LOTS X with a point-countable base and having the property that every pairwise disjoint collection of convex open sets is \sigma-discrete, and yet X does not have a \sigma-closed-discrete dense subset;
  4. there is a dense-in-itself LOTS Y with a point-countable base that does not have a \sigma-closed-discrete dense subset, and yet every nowhere dense subspace of Y has a \sigma-closed-discrete dense subset for its relative topology.

One old question [BL1] about perfect GO-spaces remains open: Is it true that every perfect GO-space can be topologically embedded in some perfect LOTS? That question is linked to Theorems 2 and 3 because W. Shi [Shi1] proved that any perfect GO-space with a \sigma-closed-discrete dense set can be embedded in a perfect LOTS.


A third major advance is due to Gruenhage, Hattori, and Ohta [GHO] who solved an old problem of Heath and Lutzer [HLu] about Dugundji extension theory.2 A special case of Theorem 1 in [GHO] is:

4. Theorem: Suppose X is a perfect GO-space whose cardinality is non-measurable, and let A be a closed subspace of X. Then the following are equivalent:

  1. there is a continuous linear extender from C(A) to C(X) where both function spaces carry the topology of pointwise convergence or both carry the compact-open topology;
  2. there is a continuous linear extender from C*(A) to C*(X) where both function spaces carry the topology of pointwise convergence or both carry the compact-open topology;
  3. there is a linear extender E: C(A) --> C(X) such that for each f in C(A), the range of E(f) is contained in the closed convex hull of the range of f;
  4. there is a linear extender E: C(A) --> C(X) such that for each f in C(A), the range of E(f) is contained in the convex hull of the range of f;
  5. for each space Y, A×Y is C*-embedded in X ×Y.
If, in addition, X is zero-dimensional, then each of the above is equivalent to
  1. A is a retract of X.

    As it happens, van Douwen [vD] had constructed a zero-dimensional separable (and hence perfect) LOTS X having a closed subset A that is not a retract of X, and had asked whether that space might provide a negative answer to the question of [HLu]. In the light of Theorem 4 above, van Douwen's example answers the question. In addition, [GHO] constructs an easier example.

    A somewhat more technical question due to the authors [BL1] was also solved recently. We asked whether a GO-space X must be quasi-developable provided each subspace of X has a \sigma-minimal base for its relative topology, and whether a compact LOTS Y must be metrizable if each of its subspaces has a \sigma-minimal base. In [BL2] we constructed a perfect LOTS E(Y,X) that is paracompact and Cech-complete, has a small diagonal and weight \omega1, and is not quasi-developable, even though each of its subspaces has a \sigma-minimal base. Shi [Shi2] solved the second and harder problem by constructing a suitable compact LOTS starting with the branch space of an Aronszajn tree.

    Other recent advances have shed light on the gap between GO-spaces with a point-countable base and GO-spaces with a \sigma-disjoint base, and on the gap between GO-spaces with a \sigma-disjoint base and those with a \sigma-discrete base (i.e., metrizable GO-spaces). Here are two examples.

    5. Theorem: [BL3] Let X be a GO-space. Then the following assertions are equivalent:

    1. X is quasi-developable;
    2. X has a \sigma-disjoint base;
    3. X has a point-countable base and has a sequence < (Un, Dn) > where for each n, Un is open in X and Dn is a relatively closed-discrete subset of Un, and if p is a point of an open set G, then for some n, p is in Un and the intersection of G and Dn is non-void.

    6. Theorem: [BL4] A GO-space X has a weakly uniform base in the sense of [HLi] if and only if X has a \sigma-disjoint base and a G\delta-diagonal.

    Other recent work has been inspired by Gruenhage's metrization theorem for compact Hausdorff spaces using properties of X2 - \Delta [Gr1] and Stepanova's metrization theorem for paracompact p-spaces [Stp1], [Stp2]. See, for example, [BL5] and [BL6].

    Finally, there is a developing new research area that involves topological properties of product spaces where one (or more) factor space is a GO-space, or even a space of ordinals. See, for example, [FS], [GNP], and [KS].

    Readers seeking more information about these topics should consult [BL7].

    References

    [BL1]
    Bennett, H. and Lutzer, D., Problems in perfect ordered spaces, in J. van Mill and G. Reed, eds, Open Problems in Topology (North Holland, Amsterdam, 1990), 223-236.
    [BL2]
    Bennett, H. and Lutzer, D., A metric space of A.H. Stone and an example concerning \sigma-minimal bases, Proc. Amer. Math. Soc., 126(1998), 2191-2196.
    [BL3]
    Bennett, H. and Lutzer, D., Point-countability in generalized ordered spaces, Topology and its Appl. 71(1996), 149-165.
    [BL4]
    Bennett, H. and Lutzer, D., Ordered spaces with special bases, Fund. Math. 158(1998), 289-299.
    [BL5]
    Bennett, H. and Lutzer, D., Off diagonal metrization theorems, Topology Proceedings 22 (1997), 37-58. http://at.yorku.ca/b/a/a/i/18.htm
    [BL6]
    Bennett, H. and Lutzer, D., Ordered spaces with continuous separating families, to appear.
    [BL7]
    Bennett, H. and Lutzer, D., Recent developments in the topology of ordered spaces, to appear.
    [vD]
    van Douwen, E., Simultaneous Extensions of Continuous Functions, Ph.D. Thesis, Vrije Universiteit, Amsterdam, 1975.
    [FS]
    Fleissner, W. and Stanley, A., D-spaces, Topology Appl., to appear.
    [Gr1]
    Gruenhage, G., Covering properties of X2 - \Delta and compact subsets of \sigma-products, Topology and its Appl. 28 (1984), 287-304
    [GHO]
    Gruenhage, G., Hattori, Y., and Ohta, H., Dugundji extenders and retracts in generalized ordered spaces, Fund. Math. 158(1998), 147-164.
    [GNP]
    Gruenhage, G., Nogura, T., and Purisch, S., Normality of X×\omega1, Topology Appl. 39(1991), 263-275.
    [HLi]
    Heath, R. and Lindgren, W., Weakly uniform bases, Houston J. Math. 2(1976), 85-90.
    [HLu]
    Heath, R. and Lutzer, D., Dugundji extension theorems for linearly ordered spaces, Pacific J. Math. 55(1974), 419-425.
    [KS]
    Kemoto, K. and Smith, K., Hereditary countable metacompactness in finite and infinite product spaces of ordinals, Topology Appl. 77(1997), 57-63.
    [QT]
    Qiao, Y-Q. and Tall, F., Perfectly normal non-metrizable non-archimedean spaces are generalized Souslin lines, Proc. Amer. Math. Soc., to appear
    [Ru1]
    Rudin, M., Compact, separable, linearly ordered spaces, Topology Appl. 82 (1998), 397-419.
    [Ru2]
    Rudin, M., Zero-dimensionality and monotone normality, Topology Appl. 85(1998), 319-333.
    [Ru3]
    Rudin, M., Nikiel's conjecture, to appear. http://at.yorku.ca/v/a/a/a/67.htm
    [Shi1]
    Shi, W., Extensions of perfect GO-spaces with \sigma-discrete dense sets, Proc. Amer. Math. Soc. 127(1999), 615-618.
    [Shi2]
    Shi, W., A non-metrizable compact LOTS each subspace of which has a \sigma-minimal base, Proc. Amer. Math. Soc. 127(1999), 2783-2791.
    [Stp1]
    Stepanova, E., Extension of continuous functions and metrizability of paracompact p-spaces, Mathematical Notes 53(1993), 308-314.
    [Stp2]
    Stepanova, E., On metrizability of paracompact p-spaces, Moscow University Mathematics Bulletin 49 (1994), 41-43.
    Harold R Bennett
    Texas Tech University, Lubbock, TX 79409

    David J. Lutzer
    College of William and Mary, Williamsburg, VA 23187

    Footnotes

    1
    Recall that a generalized ordered space (or GO-space) is a Hausdorff space X equipped with a linear ordering such that the topology has an open base consisting of order convex sets. If the topology coincides with the open interval topology of the order, then X is a linearly ordered topological space (LOTS). It is known that the class of GO-spaces is topologically the same as the class of subspaces of LOTS.
    2
    Recall that for any space Y, C(Y) (respectively C*(Y)) is the vector space of continuous (respectively, continuous and bounded) real-valued functions on Y and that for a subset A of Y, a function E: C(A) --> C(Y) is an extender provided E(f) extends f for each f in C(A). [HLu] proved that for every closed subspace A of a GO-space X, there is a linear extender E: C*(A) --> C*(X) such that the range of E(f) is always contained in the closed convex hull of the range of f, and asked whether in a perfect GO-space one could find a linear extender from C(A) to C(X) with the same range restriction.

    Date published: June 26, 2000.

    Copyright 2000 © Topology Atlas.