A note on D-spaces

We introduce notions of nearly good relations and N-sticky modulo a relation as tools for proving that spaces are D-spaces. As a corollary to general results about such relations, we show that C_p(X) is hereditarily a D-space whenever X is a Lindel\"of \Sigma-space. This answers a question of Matveev, and improves a result of Buzyakova, who proved the same result for X compact. We also prove that if a space X is the union of finitely many D-spaces, and has countable extent, then X is linearly Lindel\"of. It follows that if X is in addition countably compact, then X must be compact. We also show that Corson compact spaces are hereditarily D-spaces. These last two results answer recent questions of Arhangel'skii. Finally, we answer a question of van Douwen by showing that a perfectly normal collectionwise-normal non-paracompact space constructed by R. Pol is a D-space.Comment: 11 page

The basis problem for subspaces of monotonically normal compacta

We prove, assuming Souslin's Hypothesis, that each uncountable subspace of each zero-dimensional monotonically normal compact space contains an uncountable subset of the real line with either the metric, the Sorgenfrey, or the discrete topology.Comment: 12 page

Monotonically monolithic spaces, Corson compacts, and D-spaces

AbstractMonotonically monolithic spaces were recently introduced by V.V. Tkachuk, and monotonically κ-monolithic spaces by O. Alas, V.V. Tkachuk, and R. Wilson. In this note we answer some of their questions by showing that monotonically ω-monolithic compact spaces must be Corson compact, yet there is a Corson compact space which is not monotonically ω-monolithic. We obtain a characterization of monotonic monolithity that shows its close relationship to condition (G) of P. Collins and R. Roscoe. We also give an easy proof of Tkachukʼs result that monotonically monolithic spaces are hereditarily D-spaces by applying a result involving nearly good relations, and finally, we generalize nearly good to nearly OK to similarly obtain L.-X. Pengʼs result that weakly monotonically monolithic spaces are D-spaces

On The Decomposition of Order-separable Posets of Countable Width into Chains

partially ordered set X has countable width if and only if every collection of pairwise incomparable elements of X is countable. It is order-separable if and only if there is a countable subset D of X such that whenever p, q ∈ X and p \u3c q, there is r ∈ D such that p ≤ r ≤ q. Can every order-separable poset of countable width be written as the union of a countable number of chains? We show that the answer to this question is no if there is a 2-entangled subset of IR, and yes under the Open Coloring Axiom

A game and its relation to netweight and D-spaces

summary:We introduce a two player topological game and study the relationship of the existence of winning strategies to base properties and covering properties of the underlying space. The existence of a winning strategy for one of the players is conjectured to be equivalent to the space have countable network weight. In addition, connections to the class of D-spaces and the class of hereditarily Lindelöf spaces are shown

Baireness of Ck(X)C_k(X) for ordered XX

summary:We show that if $X$ is a subspace of a linearly ordered space, then $C_k(X)$ is a Baire space if and only if $C_k(X)$ is Choquet iff $X$ has the Moving Off Property