The notion of quasi-p-boundedness for p ∈ is introduced and investigated. We characterize quasi-p-pseudocompact subsets of β(ω) containing ω, and we show that the concepts of RK-compatible ultrafilter and P-point in can be defined in terms of quasi-p-pseudocompactness. For p ∈ , we prove that a subset B of a space X is quasi-p-bounded in X if and only if B × is bounded in X × , if and only if , where is the set of Rudin-Keisler predecessors of p.
In [7], M. Levin proved that the set of all Bing maps of a compact metric space to the unit interval is a dense -subset of the space of all maps. In [6], J. Krasinkiewicz independently proved that the set of all Bing maps of a compact metric space to an n-dimensional manifold (n ≥ 1) is a dense -subset of the space of maps. In [9], J. Song and E. D. Tymchatyn, solving some problems of J. Krasinkiewicz ([6]), proved that the set of all Bing maps of a compact metric space to a nondegenerate connected...
We prove that if X is a perfect finite-dimensional compactum, then for almost every continuous surjection of the Cantor set onto X, the set of points of maximal order is uncountable. Moreover, if X is a perfect compactum of positive finite dimension then for a typical parametrization of X on the Cantor set, the set of points of maximal order is homeomorphic to the product of the rationals and the Cantor set.
A characterization of the cozero elements of a frame, without reference to the reals, is given and is used to obtain a characterization of pseudocompactness also independent of the reals. Applications are made to the congruence frame of a -frame and to Alexandroff spaces.
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.