D-forced spaces: a new approach to resolvability

Resolvability in c.c.c. generic extensions

Every crowded space $X$ is $ω$-resolvable in the c.c.c generic extension $V^{Fn(|X|,2})$ of the ground model. We investigate what we can say about $λ$-resolvability in c.c.c-generic extensions for $λ>ω$? A topological space is "monotonically $ω_1$-resolvable" if there is a function $f:X\to {ω_1}$ such that $$\{x\in X: f(x)\ge α \}\subset^{dense}X $$ for each $α<{ω_1}$. We show that given a $T_1$ space $X$ the following statements are equivalent: (1) $X$ is $ω_1$-resolvable in some c.c.c-generic extension, (2) $X$ is monotonically $ω_1$-resolvable. (3) $X$ is $ω_1$-resolvable in the Cohen-gener…

不可数空间中的可分解性与紧性

On resolvability and tightness in uncountable spaces

本文构造实数轴上的非正则加细以揭示可分解性与无处稠密紧性的关系，并强化已有定理，得到任意基数规模的Tychonoff次极大离散紧性空间。

On the resolvability of Lindelöf-generated and (countable extent)-generated spaces

Given a topological property $P$, we say that the space $X$ is $P$-generated if for any subset $A\subset X$ that is not open in $X$ there is a subspace $Y \subset X$ with property $P$ such that $A\cap Y$ is not open in $Y$. (Of course, in this definition we could replace "open" with "closed".) In this paper we prove the following two results: (1) Every Lindelöf-generated regular space $X$ satisfying $|X|=Δ(X)=ω_1$ is $ω_1$-resolvable. (2) Any (countable extent)-generated regular space $X$ satisfying $Δ(X)>ω$ is $ω$-resolvable. These are significant strengthenings of our earlier results from [J…

Resolvability and complete accumulation points

We prove that: I. For every regular Lindelöf space $X$ if $|X|=Δ(X)$ and $\mathrm{cf}|X|\neω$, then $X$ is maximally resolvable; II. For every regular countably compact space $X$ if $|X|=Δ(X)$ and $\mathrm{cf}|X|=ω$, then $X$ is maximally resolvable. Here $Δ(X)$, the dispersion character of $X$, is the minimum cardinality of a nonempty open subset of $X$. Statements I and II are corollaries of the main result: for every regular space $X$ if $|X|=Δ(X)$ and every set $A\subseteq X$ of cardinality $\mathrm{cf}|X|$ has a complete accumulation point, then $X$ is maximally resolvable. Moreover, regu…

On resolvability of products

All spaces below are $T_0$ and crowded (i.e. have no isolated points). For $n \le ω$ let $M(n)$ be the statement that there are $n$ measurable cardinals and $Π(n)$ ($Π^+(n)$) that there are $n+1$ (0-dimensional $T_2$) spaces whose product is irresolvable. We prove that $M(1),\,Π(1)$ and $Π^+(1)$ are equiconsistent. For $1 < n < ω$ we show that $CON(M(n))$ implies $CON(Π^+(n))$. Finally, $CON(M(ω))$ implies the consistency of having infinitely many crowded 0-dimensional $T_2$-spaces such that the product of any finitely many of them is irresolvable. These settle old problems of Malychin. Concer…

Coloring Cantor sets and resolvability of pseudocompact spaces

Let us denote by $Φ(λ,μ)$ the statement that $\mathbb{B}(λ) = D(λ)^ω$, i.e. the Baire space of weight $λ$, has a coloring with $μ$ colors such that every homeomorphic copy of the Cantor set $\mathbb{C}$ in $\mathbb{B}(λ)$ picks up all the $μ$ colors. We call a space $X\,$ {\em $π$-regular} if it is Hausdorff and for every non-empty open set $U$ in $X$ there is a non-empty open set $V$ such that $\overline{V} \subset U$. We recall that a space $X$ is called {\em feebly compact} if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact iff it is feebly c…

乘积与平方空间中的可分解性

Resolvability in products and squares

本文研究拓扑空间乘积的可分解性，在基数函数与共尾性条件下证明若干极大可分解及 κ⁺-可分解结果，并给出 GCH 和 r=c 假设下的推论。

无处常值映射族与可解性

Nowhere constant families of maps and resolvability

该研究在拓扑学领域定义了无处常值映射族的概念，证明了若拓扑空间X的实连续映射族C(X)无处常值且有连通集构成的π-基，则X是c-可解的，改进了已有文献结果。

论可分解性、连通性与伪紧性

On resolvability, connectedness and pseudocompactness

本文在不同基数与集合论假设下构造次极大稠密子空间及伪紧、连通但非 $\kappa^+$-可分解的拓扑空间。

具有指定可分解性质的Tychonoff扩张

Tychonoff Expansions with Prescribed Resolvability Properties

该文利用KID扩张方法，证明任意满足特定条件的Tychonoff空间可构造出五种具有不同可分解性质的Tychonoff扩张，从而否定地回答了两个经典问题。