Richard G. Wilson

Irresolvable and submaximal spaces: Homogeneity versus σ-discreteness and new ZFC examples1

Abstract An example of an irresolvable dense subspace of {0,1} c is constructed in ZFC. We prove that there can be no dense maximal subspace in a product of first countable spaces, while under Booths Lemma there exists a dense submaximal subspace in [0,1] c . It is established that under the axiom of constructibility any submaximal Hausdorff space is σ-discrete. Hence it is consistent that there are no submaximal normal connected spaces. If there exists a measurable cardinal, then there are models of ZFC with non-σ-discrete maximal spaces. We prove that any homogeneous irresolvable space of non-measurable cardinality is of first category. In particular, any homogeneous submaximal space is strongly σ-discrete if there are no measurable cardinals.

Almost discrete SV-spaces

Abstract A Hausdorff space is called almost discrete if it has precisely one nonisolated point. A Tychonoff space Y is called an SV-space if C(Y) P is a valuation ring for every prime ideal P of C(Y) . it is shown that the almost discrete space X = D ∪{∞} is an SV-space if and only if X is a union of finitely many closed basically disconnected subspaces if and only if M ∞ ={ƒϵC(X):ƒ(∞)=0} contains only finitely many minimal prime ideals. Some unsolved problems are posed.

When Is C(X)/P a Valuation Ring for Every Prime Ideal P?

Abstract A Tychonoff space X is called an SV-space if for every prime ideal P of the ring C(X) of continuous real-valued functions on X, the ordered integral domain C (X) P is a valuation ring (i.e., of any two nonzero elements of C (X) P , one divides the other). It is shown that X is an SV-space iff υX is an SV-space iff βX is an SV-space. If every point of X has a neighborhood that is an F-space, then X is an SV-space. An example is supplied of an infinite compact SV-space such that any point with an F-space neighborhood is isolated. It is shown that the class of SV-spaces includes those Tychonoff spaces that are finite unions of C∗-embedded SV-spaces. Some open problems are posed.

On the extent of star countable spaces

For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ⊂ X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have arbitrary extent. It is proved that for any ω1-monolithic compact space X, if Cp(X)is star countable then it is Lindelöf.

Connectifying some spaces

Abstract A Hausdorff space X is called (countably) connectifiable if there exists a connected Hausdorff space Y (with |Y⊮X| ⩽ ω ; respectively) such that X embeds densely into Y . We prove that it is consistent with ZFC that there exists a regular dense in itself countable space which is not countably connectifiable giving thus a partial answer to Problem 3.9 of Watson and Wilson (1993). On the other hand we show that Martins axiom implies that every countable dense in itself space X with πω ( X ) ω is countably connectifiable. We also establish that a separable metrizable space without open compact subsets can be densely embedded in a metric continuum.

Pseudocompact Whyburn spaces need not be Fréchet

We prove in ZFC that there exists a Tychonoff pseudocompact scattered AP-space of uncountable tightness. We give some sufficient and necessary conditions for a P-space to be AP as well as a characterization of AP-property in linearly ordered topological spaces.

Neither first countable nor Cech-complete spaces are maximal Tychonoff connected

A connected Tychonoff space X is called maximal Tychonoff connected if there is no strictly finer Tychonoff connected topology on X. We show that if X is a connected Tychonoff space and X E {locally separable spaces, locally Cech-complete spaces, first countable spaces}, then X is not maximal Tychonoff connected. This result is new even in the cases where X is compact or metrizable.

Compatible connectedness in graphs and topological spaces

A topology on the vertex set of a graphG iscompatible with the graph if every induced subgraph ofG is connected if and only if its vertex set is topologically connected. In the case of locally finite graphs with a finite number of components, it was shown in [11] that a compatible topology exists if and only if the graph is a comparability graph and that all such topologies are Alexandroff. The main results of Section 1 extend these results to a much wider class of graphs. In Section 2, we obtain sufficient conditions on a graph under which all the compatible topologies are Alexandroff and in the case of bipartite graphs we show that this condition is also necessary.

Digital Jordan curves — a graph-theoretical approach to a topological theorem

Abstract We give a proof of a graph-theoretical Jordan curve theorem which generalizes both the topological results of Khalimsky et al. as well as some of the graph-theoretical results of Rosenfeld and Klette and Voss.

Maximal pseudocompact spaces

If \(\mathcal {P}\) is a topological property and \(\mathcal C\) is a class of topologies, then a space X is said to be maximal \(\mathcal {P}\) in the class \(\mathcal C\) if X has \(\mathcal {P}\) but no strictly stronger topology on X which belongs to the class \(\mathcal C\) has \(\mathcal {P}\). Recall that a topological space X (with no separation axiom assumed) is feebly compact (called lightly compact in [1]) if every locally finite family of non-empty open subsets of X is finite, or equivalently if every countable nested family of regular closed sets has non-empty intersection. It is well-known that in the class of Tychonoff spaces, feeble compactness is equivalent to pseudocompactness and hence maximal pseudocompactness is equivalent to maximal feeble compactness in the class of Tychonoff spaces.