Ideal Resolvability
by
Julian Dontchev
University of Helsinki
Coauthors: Maximilian Ganster, David Rose

In 1943, Hewitt introduced the concept of a resolvable space. By definition, a nonempty topological space (X, \tau) is called resolvable if X is the disjoint union of two dense (or equivalently codense) subsets. In the opposite case X is called irresolvable. Every space (X, \tau) has its unique Hewitt representation, i.e.  X = F \cup G, where F is closed and resolvable, G is hereditarily irresolvable and F \cap G = \emptyset.

A nonempty collection I of subsets on a topological space (X, \tau) is called a topological ideal on (X, \tau) if it satisfies the following two conditions:

(1) If A in I and B subset or equal A, then B in I (heredity).

(2) If A in I and B in I, then A \cup B in I (finite additivity).

For a space (X, \tau, I) and a subset A subset or equal X, A^*(I) = {x in X: U \cap A not in I for every U in \tau (x)} is called the local function of A with respect to I and \tau.

A subset A of a topological space (X, \tau, I) is called I-dense if every point of X is in the local function of A with respect to I and \tau, i.e.  if A^* (I) = X. A nonempty topological space (X, \tau, I) is called \mathal I-resolvable if X has two disjoint I-dense subsets.

Recall that a measurable set E subset or equal R has density d at x in R if

lim h --> 0  m(E \cap [x-h, x+h]) / 2h

exists and is equal to d. Set \phi(E) = { x in R : d(x, E) = 1 }.

The open sets of the density topology \tau_d are those measurable sets E that satisfy E subset or equal \phi(E). Clearly the density topology \tau_d is finer than the usual topology on the real line.

The aim of this paper is to study in details resolvability modulo an ideal and to prove that the density topology is (M-)resolvable. Additionally maximal I-resolvability and bounded resolvability is investigated.

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-19.