This paper relies on nested postulates of separate, linear and arc-continuity of functions to define analogous properties for sets that are weaker than the requirement that the set be open or closed. This allows three novel characterisations of open or closed sets under convexity or separate convexity postulates: the first pertains to separately convex sets, the second to convex sets and the third to arbitrary subsets of a finite-dimensional Euclidean space. By relying on these constructions, we also obtain new results on the relationship between separate and joint continuity of separately quasiconcave, or separately quasiconvex functions. We present examples to show that the sufficient conditions we offer cannot be dispensed with.

By a tiling of a topological linear space $X$, we mean a covering of $X$ by at least two closed convex sets, called tiles, whose nonempty interiors are pairwise disjoint. Study of tilings of infinite dimensional spaces was initiated in the 1980's with pioneer papers by V. Klee. We prove some general properties of tilings of locally convex spaces, and then apply these results to study the existence of tilings of normed and Banach spaces by tiles possessing certain smoothness or rotundity properties. For a Banach space $X$, our main results are the following.

• (i) $X$ admits no tiling by Fréchet smooth bounded tiles.

• (ii) If $X$ is locally uniformly rotund $\left( \text{LUR} \right)$, it does not admit any tiling by balls.

• (iii) On the other hand, some ${{\ell }_{1}}\left( \Gamma \right)$ spaces, $\Gamma $ uncountable, do admit a tiling by pairwise disjoint $\text{LUR}$ bounded tiles.

The Rank Theorem is proved for locally Lipschitz continuous functions f:Rn → Rp with generalized derivative of constant rank.

Two results are obtained in this paper. The first is a generalization of the imprimitivity theorem of Mackey, when the associated projection-valued measure is replaced by a commutative positive operator valued measure. The second is a necessary and sufficient condition for such a system of covariance to possess an overcomplete, covariant family of coherent states.