On a simple invariant of Turaev-Viro type
Let be a compact, orientable, irreducible 3-manifold with a torus. We show that there can be infinitely many slopes on realized by the boundary curves of immersed, incompressible, - incompressible surfaces in which are embedded in a neighborhood of .
Let k be a fixed natural number. We show that if C is a closed and nonconvex set in Hilbert space such that the closures of the projections onto all k-hyperplanes (planes with codimension k) are convex and proper, then C must contain a closed copy of Hilbert space. In order to prove this result we introduce for convex closed sets B the set consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that is precisely the intersection of all k-imitations...
For a stratified mapping , we consider the condition concerning the kernel of the differential of . We show that the condition is equivalent to the condition which has a more obvious geometric content.
We present an example of a connected, Polish, countable dense homogeneous space X that is not strongly locally homogeneous. In fact, a nontrivial homeomorphism of X is the identity on no nonempty open subset of X.