Lower Density Topologiesa

Abstract

By considering lower density operators and their induced topologies in a general setting, some results of S. Scheinberg and E. Lazarow et al. are unified and generalized.It is also shown that every u-finite complete measure space (X, A', m ) has a lower density operator and that every such operator induces a topology makingXa category measure space in the sense of J. C. Oxtoby, except that the measure need not be finite. One consequence is that category u-finite measure spaces must have the countable chain condition. Also, for every topological space (X, T), there is a lower density operator on the u-field of sets having the property of Baire (relative to the u-ideal of meager sets). Further, in both the "measure" and "category" contexts, all induced lower density topologies have simple form. Finally, it is shown that the deep.Pdensity operator on the u-field of subsets of the real line having the property of Baire is not a lower density operator. Mathematics Subject Classification Primary 28A05,54A10.Key words and phrases. Lower density operator, lower density topology, (u-)field, (a-)ideal, set having the property of Baire, meager set, nowhere dense set, (u-) finite measure space, category measure space, CCC (countable chain condition).

On lower density type operators and topologies generated by them

“…It turns out that operators from the family LLDO play a special role in the considerations. As we mentioned earlier in [14] one can find that for any measurable space X, S, J if LDO ∅ then LLDO ∅. We start with the following lemma useful in the next part of the paper.…”

“…Since T ∅, Theorem 2.1 implies that X, S, J has the hull property. Moreover, we have LDO ∅ and, in consequence, there is lifting Ψ on X, S, J (see [14]). Obviously, by Theorem…”

“…The implication (i) ⇒ (ii) is a consequence of Theorem 3.3. In [14] one can find that for any measurable space X, S, J if LDO ∅ then LLDO ∅.…”

“…Proof. To find a maximal element in SLDO we apply here a method described in the acknowledgment section of the paper [14]. Let Φ ∈ SLDO and let F x = {A ∈ S : x ∈ Φ(A)} for x ∈ X.…”

On spaces whose nowhere dense subsets are scattered

“…This last condition is equivalent to A being disjoint with A" (2-), where A" (2") {z 6 X" U Iq A 2" for every U 6 'x} with 7"x being the open neighborhood system at a point z 6 X. For more details concerning the last concepts we refer the reader to [4,9]. Clearly every scattered and every -space, i.e.…”