arXiv subjects
Publications and source records attributed to Ramiro de la Vega.
We investigate the question of when a topological space $X$ has the $\textit{Generalized Bolzano-Weierstrass property}$: every sequence of subsets of $X$ has a convergent subsequence (in the sense of Kuratowski).
Given a hereditarily meager ideal $\mathcal{I}$ on a countable set $X$ we use Martin's axiom for countable posets to produce a zero-dimensional maximal topology $τ^\mathcal{I}$ on $X$ such that $τ^\mathcal{I}\cap \mathcal{I}=\{\emptyset\}$ and, moreover, if $\mathcal{I}$ is $p^+$ then $τ^\mathcal{I}$ is selectively separable (SS) and if $\mathcal{I}$ is $q^+$, so is $τ^\mathcal{I}$. In particular, we obtain regular maximal spaces satisfying all boolean combinations of the properties SS and $q^+$.
We study the question of when an uncountable ccc topological space $X$ contains a ccc subspace of size $\aleph_1$. We show that it does if $X$ is compact Hausdorff and more generally if $X$ is Hausdorff with $\mathrm{pct}(X) \leq \aleph_1$. For each regular cardinal $κ$, an example is constructed of a ccc Tychonoff space of size $κ$ and countable pseudocharacter but with no ccc subspace of size less than $κ$. We also give a ccc compact $T_1$ space of size $κ$ with no ccc subspace of size less than $κ$.
A structure $\mathcal{A}=\left(A;E_i\right)_{i\in n}$ where each $E_i$ is an equivalence relation on $A$ is called an $n$-grid if any two equivalence classes coming from distinct $E_i$'s intersect in a finite set. A function $χ: A \to n$ is an acceptable coloring if for all $i \in n$, the set $χ^{-1}(i)$ intersects each $E_i$-equivalence class in a finite set. If $B$ is a set, then the $n$-cube $B^n$ may be seen as an $n$-grid, where the equivalence classes of $E_i$ are the lines parallel to the $i$-th coordinate axis. We use elementary submodels of the universe to characterize those $n$-grids which admit an acceptable coloring. As an application we show that if an $n$-grid $\mathcal{A}$ does not admit an acceptable coloring, then every finite $n$-cube is embeddable in $\mathcal{A}$.
We show (in ZFC) that the cardinality of a compact homogeneous space of countable tightness is no more than the size of the continuum.
Under Jensen's Diamond Principle, we show how to construct a large compact S-space while having some control over its group of autohomeomorphisms. In particular we can make the space rigid or h-homogeneous (i.e. any two clopen subsets are homeomorphic). We also get a space which is B-homogeneous and not h-homogeneous, partially answering a question of M.V. Matveev. A space is B-homogeneous if it has a base every element of which can be mapped onto every other one by a homeomorphism of the whole space.
Under the continuum hypothesis, there is a compact homogeneous strong S-space.