We prove, among other things, that if $C_p(X)$ is subcompact in the sense of de Groot, then the space $X$ is discrete. This generalizes a series of previous results on completeness properties of function spaces.

A space $X$ is a $D$-space if, for every neighborhood assignment $f$ there is a closed discrete set $D$ such that $\bigcup{f(D)}=X$. In this paper we give some necessary conditions and some sufficient conditions for a resolution of a topological space to be a $D$-space. In particular, if a space $X$ is resolved at each $x\in X$ into a $D$-space $Y_x$ by continuous mappings $f_x\colon X-\{{x}\} \rightarrow Y_x$, then the resolution is a $D$-space if and only if $\bigcup{\{{x}\}}\times \Bd(Y_x)$ is a $D$-space.

If ${\cal A} $ and ${\cal B}$ are disjoint ideals on $\omega$, there is a {\it tower preserving\/} $\sigma$-centered forcing which introduces a subset of $\omega$ which meets every infinite member of ${\cal A}$ in an infinite set and is almost disjoint from every member of ${\cal B}$. We can then produce a model in which all compact separable radial spaces are Fr\'echet, thus answering a question of P.~Nyikos. The question of the existence of compact ccc radial spaces which are not Fr\'echet was first asked by Chertanov (see \cite{Ar78}).