Dense families of selections and finite-dimensional spaces

Diagonalizations of dense families

We develop a unified framework for the study of properties involving diagonalizations of dense families in topological spaces. We provide complete classification of these properties. Our classification draws upon a large number of methods and constructions scattered in the literature, and on some novel results concerning the classical properties.

Dense families of countable sets below $c$

We show that it is consistent that the continuum is as large as you wish, and for each uncountable cardinal $κ$ below the continuum, there are a subset $T$ of the reals and a family $A$ of countable subsets of $T$ such that (1) both $T$ and $A$ have cardinality $κ$, (2) $|\bar{a}\cap T|=κ$ for each $a\in A$, (3) for each uncountable subset of $T$ contains some elements of $A$, and so (i) there is an almost disjoint family of subsets of the reals with size and chromatic number $κ$, (ii) there is a locally compact, locally countable $T_2$ space with cardinality spectrum $\{ω,κ\}$.

Selectors for dense subsets of function spaces

Let $USC^*_p(X)$ be the topological space of real upper semicontinuous bounded functions defined on $X$ with the subspace topology of the product topology on ${}^X\mathbb{R}$. $\tildeΦ^{\uparrow},\tildeΨ^{\uparrow}$ are the sets of all upper sequentially dense, upper dense or pointwise dense subsets of $USC^*_p(X)$, respectively. We prove several equivalent assertions to the assertion $USC^*_p(X)$ satisfies the selection principles $S_1(\tildeΦ^{\uparrow},\tildeΨ^{\uparrow})$, including a condition on the topological space $X$. We prove similar results for the topological space $C^*_p(X)$ of c…

Classification of selectors for sequences of dense sets of Cp(X)

We give the characteristics of selectors for sequences of dense sets of Cp(X) through the selection principles of a Tychonoff space X.

Denumerable cellular families in Hausdorff spaces and towers of Boolean algebras in $\mathbf{ZF}$

A denumerable cellular family of a topological space $\mathbf{X}$ is an infinitely countable collection of pairwise disjoint non-empty open sets of $ \mathbf{X}$. It is proved that the following statements are equivalent in $\mathbf{ZF}$: (i) For every infinite set $X,[X]^{<ω}\mathbf{\ }$has a denumerable subset. (ii) Every infinite $0$-dimensional Hausdorff space admits a denumerable cellular family. It is also proved that (i) implies the following: (iii) Every infinite Hausdorff Baire space has a denumerable cellular family. Among other results, the following theorems are also proved in $\ma…

极不连通空间的两个选择定理

Two Selection Theorems for Extremally Disconnected Spaces

本文给出极不连通空间上usco映射连续选择存在性的Hasumi定理及稠密定义连续选择延拓结果的简洁证明，并推广至正则值域。