We simultaneously generalize Silver’s perfect set theorem for co-analytic equivalence relations and Harrington-Marker-Shelah’s Dilworth-style perfect set theorem for Borel quasi-orders, establish the analogous theorem at the next definable cardinal, and give further generalizations under weaker definability conditions.

We study some geometric properties related to the set

$${{\Pi }_{X}}\,:=\left\{ \left( x,\,{{x}^{*}} \right)\,\in \,{{\text{S}}_{X}}\,\times \,{{\text{S}}_{{{X}^{*}}}}\,:\,{{x}^{*}}\left( x \right)\,=\,1 \right\}$$

obtaining two characterizations of Hilbert spaces in the category of Banach spaces. We also compute the distance of a generic element $\left( h,\,k \right)\,\in \,H\,{{\oplus }_{2}}\,H$ to ${{\Pi }_{H}}$ for $H$ a Hilbert space.

In the paper we formulate a Covering Property Axiom, $\text{CP}{{\text{A}}_{\text{prism}}}$, which holds in the iterated perfect set model, and show that it implies the following facts, of which (a) and (b) are the generalizations of results of J. Steprāns.

(a) There exists a family $\mathcal{F}$ of less than continuum many ${{C}^{1}}$ functions from $\mathbb{R}$ to $\mathbb{R}$ such that ${{\mathbb{R}}^{2}}$ is covered by functions from $\mathcal{F}$, in the sense that for every $\left\langle x,\,y \right\rangle \,\in \,{{\mathbb{R}}^{2}}$${{\mathbb{R}}^{2}}$ there exists an $f\,\in \,\mathcal{F}$ such that either $f\left( x \right)\,=\,y$ or $f\left( y \right)\,=\,x$.

(b) For every Borel function $f:\,\mathbb{R}\,\to \,\mathbb{R}$ there exists a family $\mathcal{F}$ of less than continuum many “${{C}^{1}}$” functions (i.e., differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of $f$.

(c) For every $n\,>\,0$ and a ${{D}^{n}}$ function $f:\,\mathbb{R}\,\to \,\mathbb{R}$ there exists a family $\mathcal{F}$ of less than continuum many ${{C}^{n}}$ functions whose graphs cover the graph of $f$.

We also provide the examples showing that in the above properties the smoothness conditions are the best possible. Parts (b), (c), and the examples are closely related to work of A. Olevskiî.