Cardinal invariants related to sequential separability of generalized Cantor cubes 2κ, introduced by M. Matveev, are studied here. In particular, it is shown that the following assertions are relatively consistent with ZFC: (1) 2ω1 is sequentially separable, yet there is a countable dense subset of 2ω1 containing no non-trivial convergent subsequence, (2) 2ω1 is not sequentially separable, yet ...

Ergodic Optimization is the process of finding invariant probability measures that maximize the integral of a given function. It has been conjectured that “most” functions are optimized by measures supported on a periodic orbit, and it has been proved in several separable spaces that an open and dense subset of functions is optimized by measures supported on a periodic orbit. All known positive...

Notes on set theory, mainly forcing. The first four sections closely follow the lecture notes Williams [8] and the book Kunen [4]. The last section covers topics from various sources, as indicated there. Hopefully, all errors are mine. 1. The Suslin problem 1.1. The Suslin hypothesis. Recall that R is the unique dense, complete and separable order without endpoints (up to isomorphism). It follo...

Definition 1.1. Let (X, τ) be a topological space. A subset D ⊆ X is called dense if D ∩O 6= ∅ for every nonempty open set O ⊆ X. X is called separable if X has a countable dense subset. X is called metrizable if there is a metric d on X such that the topology τ is the same as the topology induced by the metric. The metric is called complete if every Cauchy sequence converges in X. Finally, X i...

Rudin, Chapter 4, Problem #3. The space L(T ) is separable since the trigonometric polynomials with complex coefficients whose real and imaginary parts are rational form a countable dense subset. (Denseness follows from Theorem 3.14 and Theorem 4.25; countability is clear since {e | n ∈ Z} is a countable basis of the trigonometric polynomials.) Meanwhile the space L(T ) is not separable. To see...

We investigate the relations of almost isometric embedding and of almost isometry between metric spaces. These relations have several appealing features. For example, all isomorphism types of countable dense subsets of R form exactly one almost-isometry class, and similarly with countable dense subsets of Uryson’s universal separable metric space U. We investigate geometric, set-theoretic and m...

We study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely n types of countable dense sets: such a space contains a subset S of size at most n−1 such that S is invariant under all homeomorphisms of X and X \ S is countable dense homogeneous. We prove that every Borel space having fewer than c types of count...

In these notes we provide a new proof of the existence of a hypercyclic uniformly continuous semigroup of operators on any separable infinitedimensional Banach space that is very different from—and considerably shorter than—the one recently given by Bermúdez, Bonilla and Martinón. We also show the existence of a strongly dense family of topologically mixing operators on every separable infinite...