We study countable splitting of Markushevich bases in weakly Lindel\"of Banach spaces in connection with the geometry of these spaces.

Certain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees.

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.