As a special case of the general question - “What information can be obtained about the dimension of a subset of by looking at its orthogonal projections into hyperplanes?” - we construct a Cantor set in each of whose projections into 2-planes is 1-dimensional. We also consider projections of Cantor sets in whose images contain open sets, expanding on a result of Borsuk.
Let X be a set with a symmetric kernel d (not necessarily a distance). The space (X,d) is said to have the weak (resp. strong) covering property of degree ≤ m [briefly prf(m) (resp. prF(m))], if, for each family B of closed balls of (X,d) with radii in a decreasing sequence (resp. with bounded radii), there is a subfamily, covering the center of each element of B, and of order ≤ m (resp. splitting into m disjoint families). Since Besicovitch, covering properties are known to be the main tool for...
We study some relations whose compatibility with the topology is equivalent to normality or to complete regularity.
We give some necessary and sufficient conditions for the Scott topology on a complete lattice to be sober, and a sufficient condition for the weak topology on a poset to be sober. These generalize the corresponding results in [1], [2] and [4].