Related papers: Completely Baire spaces, Menger spaces, and projec…
Hurewicz proved completely metrizable Menger spaces are /sigma-compact. We extend this to Cech-complete Menger spaces and consistently to projective Menger metrizable spaces. On the other hand, it is consistent that there is a co-analytic…
Menger's conjecture that Menger spaces are /sigma-compact is false; it is true for analytic subspaces of Polish spaces and undecidable for more complex definable subspaces of Polish spaces. For non-metrizable spaces, analytic Menger spaces…
We introduce a new topological generalization of the $\sigma$-projective hierarchy, not limited to Polish spaces. Earlier attempts have replaced $^{\omega}\omega$ by $^{\kappa}\kappa$, for $\kappa$ regular uncountable, or replaced countable…
All spaces are assumed to be separable and metrizable. Consider the following properties of a space $X$. (1) $X$ is Polish. (2) For every countable crowded $Q\subseteq X$ there exists a crowded $Q'\subseteq Q$ with compact closure. (3)…
The main result is the following. Let $f \colon X \rightarrow Y$ be a continuous mapping of a completely Baire space $X$ onto a hereditary weakly Preiss-Simon regular space $Y$ such that the image of every open subset of $X$ is a resolvable…
We prove that, for every n, the topological space {\omega}_n^{\omega} (where {\omega}_n has the discrete topology) can be partitioned into {\omega}_n copies of the Baire space. Using this fact, the authors then prove two new theorems about…
We show that the set of codes for Ramsey positive analytic sets is $\mathbf{\Sigma}^1_2$-complete. This is a one projective-step higher analogue of the Hurewicz theorem saying that the set of codes for uncountable analytic sets is…
Menger's basis property is a generalization of $\sigma$-compactness and admits an elegant combinatorial interpretation. We introduce a general combinatorial method to construct non $\sigma$-compact sets of reals with Menger's property.…
The space of unitary $C_{0}$-semigroups on separable infinite dimensional Hilbert space, when viewed under the topology of uniform weak convergence on compact subsets of $\mathbb{R}_{+}$, is known to admit various interesting residual…
For a metrizable space, we consider the space of all metrics generating the same topology of the metrizable space, and this space of metrics is equipped with the supremum metric. In this paper, for every metrizable space, we establish that…
We prove several theorems on sigma-bounded and sigma-compact pointsets. We start with a known theorem by Kechris, saying that any lightface \Sigma^1_1 set of the Baire space either is effectively sigma-bounded (that is, covered by a…
We provide new techniques to construct sets of reals without perfect subsets and with the Hurewicz or Menger covering properties. In particular, we show that if the Continuum Hypothesis holds, then there are such sets which can be mapped…
Working over infinite dimensional separable Hilbert spaces, residual results have been achieved for the space of contractive $C_{0}$-semigroups under the topology of uniform weak operator convergence on compact subsets of $\mathbb{R}_{+}$.…
All spaces are assumed to be separable and metrizable. Our main result is that the statement "For every space $X$, every closed subset of $X$ has the perfect set property if and only if every analytic subset of $X$ has the perfect set…
We provide simplified solutions of Menger's and Hurewicz's problems and conjectures, concerning generalizations of sigma-compactness. The reader who is new to this field will find a self-contained treatment in Sections 1, 2, and 5. Sections…
We study `definable' subsets of Baire space $\mathcal{N}$. The logic of our arguments is intuitionistic and we use L.E.J.~Brouwer's Thesis on bars in $\mathcal{N}$ and his continuity axioms. We avoid the operation of taking the complement…
Menger conjectured that subsets of $\mathbb R$ with the Menger property must be $\sigma$-compact. While this is false when there is no restriction on the subsets of $\mathbb R$, for projective subsets it is known to follow from the Axiom of…
We prove that it is consistent (relative to a Mahlo cardinal) that all projective sets of reals are Lebesgue measurable, but there is a $\Delta^1_3$ set without the Baire property. The complexity of the set which provides a counterexample…
The usual definition of the set of constructible reals is $\Sigma ^1_2$. This set can have a simpler definition if, for example, it is countable or if every real is constructible. H. Friedman asked if the set of constructible reals can be…
Given a metrizable space $X$, let $AM(X)$ be the space of continuous bounded admissible metrics on $X$, which is endowed with the sup-metric. In this paper, we shall investigate the Borel complexity and the complete metrizability of $AM(X)$…