Properties of Lipschitz and d.c. surfaces of finite codimension in a Banach space and properties of generated $\sigma$-ideals are studied. These $\sigma$-ideals naturally appear in the differentiation theory and in the abstract approximation theory. Using these properties, we improve an unpublished result of M. Heisler which gives an alternative proof of a result of D. Preiss on singular points of convex functions.

Albeverio, Kondratiev, and Röckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space ${\Gamma }_X$ of any manifold $X$. The name comes from the fact that various elements of the geometry of ${\Gamma }_X$ are constructed via lifting of the corresponding elements of the geometry of $X$. In this note, we construct a general algebraic framework for lifted geometry which can be applied to various “infinite dimensional spaces” associated to $X$. In order to define a lifted...

Let F = ind lim Fₙ be an infinite-dimensional LF-space with density dens F = τ ( ≥ ℵ ₀) such that some Fₙ is infinite-dimensional and dens Fₙ = τ. It is proved that every open subset of F is homeomorphic to the product of an ℓ₂(τ)-manifold and ${ℝ}^{\infty }=indlimℝⁿ$ (hence the product of an open subset of ℓ₂(τ) and ${ℝ}^{\infty }$). As a consequence, any two open sets in F are homeomorphic if they have the same homotopy type.