Semilattice structures on trees.
Semi-radiality in products.
Separable extensions of first countable spaces
Sequential compactness vs. countable compactness
The general question of when a countably compact topological space is sequentially compact, or has a nontrivial convergent sequence, is studied from the viewpoint of basic cardinal invariants and small uncountable cardinals. It is shown that the small uncountable cardinal 𝔥 is both the least cardinality and the least net weight of a countably compact space that is not sequentially compact, and that it is also the least hereditary Lindelöf degree in most published models. Similar results, some definitive,...
Sequential completeness and -sequential completeness are different
Sequential + separable vs sequentially separable and another variation on selective separability
A space X is sequentially separable if there is a countable D ⊂ X such that every point of X is the limit of a sequence of points from D. Neither “sequential + separable” nor “sequentially separable” implies the other. Some examples of this are presented and some conditions under which one of the two implies the other are discussed. A selective version of sequential separability is also considered.
Short branches in Rudin-Frolík order
Short branches in the Rudin-Frolík order
Some applications of the point-open subbase game
Given a subbase of a space , the game is defined for two players and who respectively pick, at the -th move, a point and a set such that . The game stops after the moves have been made and the player wins if ; otherwise is the winner. Since is an evident modification of the well-known point-open game , the primary line of research is to describe the relationship between and for a given subbase . It turns out that, for any subbase , the player has a winning strategy...
Some applications of tiny sequences
Some Baire category type theorems for
Some combinatorial properties of ultrafilters
Some combinatorics involving ultrafilters
Some conditions under which a uniform space is fine
Let be a uniform space of uniform weight . It is shown that if every open covering, of power at most , is uniform, then is fine. Furthermore, an -metric space is fine, provided that every finite open covering is uniform.
Some factorization theorems for closed subspaces
Some Generalizations of Lindelöf and Paracompact Spaces.
Some new cardinal inequalities involving a cardinal function less than the spread and the density
Some non-multiplicative properties are -invariant
A cardinal function (or a property ) is called -invariant if for any Tychonoff spaces and with and linearly homeomorphic we have (or the space has () iff ). We prove that the hereditary Lindelöf number is -invariant as well as that there are models of in which hereditary separability is -invariant.
Some non-normal subspaces of the Čech-Stone compactification of a discrete space
Some nowhere densely generated topological properties