Let X be a topological vector space, Y = IR n , n 2 IN, A a continuous linear map from X to Y , C X, B a convex set dense in C, and d 2 Y a data point. We derive conditions which guarantee that the set B \ A ?1 (d) is nonempty and dense in C \ A ?1 (d). Some applications to shape preserving interpolation and approximation are described.

It is well-known that if T is an everywhere defined bounded operator on a Banach space X to a Banach space Y and T* is its adjoint, then the range R(T*) of T* is closed in X* if and only if the range R(T) of T is closed in Y (cf. [2, pp. 487-489]). The object of this note is to establish this result for closed but possibly unbounded operators. This result, for the unbounded case, is of great ut...

In this paper, we study dense subsets of boundaries of CAT(0) groups. Suppose that a group G acts geometrically on a CAT(0) space X and suppose that there exists an element g0 ∈ G such that (1) Zg0 is finite, (2) X \ Fg0 is not connnected, and (3) each component of X \ Fg0 is convex and not g0-invariant, where Zg0 is the centralizer of g0 and Fg0 is the fixed-point set of g0 in X (that is, Zg0 ...

We show that the 0 2 enumeration degrees are not dense. This answers a question posed by Cooper. 1 Background Cooper 3] showed that the enumeration degrees (e-degrees) are not dense. He also showed that the e-degrees of the 0 2 sets are dense 2]. Since Cooper's nondensity proof constructs a 0 7 set, he posed the question: What is the least n (2 n 6) such that the e-degrees below 0 n are not den...