相关论文: Compact Scattered Spaces in Forcing Extensions

An $\omega_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $\omega_1$-compact space is $\sigma$-countably compact, i.e., the union of…

We show that there is a compact topological space carrying a measure which is not a weak* limit of finitely supported measures but is in the sequential closure of the set of such measures. We construct compact spaces with measures of…

We show (in ZFC) that the cardinality of a compact homogeneous space of countable tightness is no more than the size of the continuum.

We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. (2) There may exist fewer than continuum…

The {\em Singular Cardinal Hypothesis} (SCH) is one of the most classical combinatorial principles in set theory. It says that if $\kappa$ is singular strong limit, then $2^{\kappa}=\kappa^+$. We prove that given a singular cardinal…

We answer a question of Moore by building a forcing extension satisfying measuring together with CH. The construction works over any model of ZFC and can be described as a forcing iteration with countable structures as side conditions and…

We examine dimensional types of scattered $P$-spaces of weight $\omega_1$. Such spaces can be embedded into $\omega_2$. There are established similarities between dimensional types of scattered separable metric spaces and dimensional types…

We give a combinatorial characterization of countable submaximal subspaces of $2^\kappa$. Using a parametrized version of Mathias forcing, we prove that there exists a countable submaximal subspace of $2^{\omega_1}$ whilst…

Combining stationary reflection (a compactness property) with the failure of SCH (an instance of non-compactness) has been a long-standing theme. We obtain this at $\aleph_{\omega_1}$, answering a question of Ben-Neria, Hayut, and Unger: We…

A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties…

It is shown that CH implies the existence of a compact Hausdorff space that is countable dense homogeneous, crowded and does not contain topological copies of the Cantor set. This contrasts with a previous result by the author which says…

The categoricity spectrum of a class of structures is the collection of cardinals in which the class has a single model up to isomorphism. Assuming that cardinal exponentiation is injective (a weakening of the generalized continuum…

If T has only countably many complete types, yet has a type of infinite multiplicity then there is a ccc forcing notion Q such that, in any Q --generic extension of the universe, there are non-isomorphic models M_1 and M_2 of T that can be…

In this paper, we prove that in forcing extensions by a poset with finally property K over a model of GCH+$\square$, every compact sequentially compact space is weakly pseudoradial. We also prove the following assuming $\mathfrak{s}\leq…

This paper presents the main results in my Ph.D. thesis. In what follows several proofs of SCH are presented introducing a family of covering properties which implies both SCH and the failure of various forms of square. These covering…

A space $X$ is called {\it selectively pseudocompact} if for each sequence $(U_{n})_{n\in \mathbb{N}}$ of pairwise disjoint nonempty open subsets of $X$ there is a sequence $(x_{n})_{n\in \mathbb{N}}$ of points in $X$ such that $cl_X(\{x_n…

We propose a reinterpretation of the continuum grounded in the stratified structure of definability rather than classical cardinality. In this framework, a real number is not an abstract point on the number line, but an object expressible…

We produce a model of ZFC in which there are no locally compact first countable S-spaces, and in which 2^{aleph_0}<2^{aleph_1}. A consequence of this is that in this model there are no locally compact, separable, hereditarily normal spaces…

Given a compact space in a fixed universe of set theory, one can naturally define its interpretation in any ZFC extension of the universe. We investigate the stability of some classes of compact spaces with respect to extensions of this…

We show that the forcing axiom for countably compact, $\omega_2$-Knaster, well-met posets is inconsistent. This is supplemental to an inconsistency result of Shelah and sets a new limit to the generalization of Martin's Axiom to the stage…