Combinatorial properties on nodec countable spaces with analytic topology

Abstract

We study some variations of the product topology on families of clopen subsets of 2 N × N in order to construct countable nodec regular spaces (i.e. in which every nowhere dense set is closed) with analytic topology which in addition are not selectively separable and do not satisfy the combinatorial principle q + .

Selective separability and $q^+$ on maximal spaces

“…Every countable p − regular space is selectively separable and discretely generated (see [6,11]). In summary, we have the following implications for countable regular spaces.…”

“…As we have already mentioned, analytic nodec spaces are hard to define and they are the only examples we know of non DG analytic spaces. In [11] was constructed an analytic regular space Y(I * * nd ) which is nodec, non q + and non SS. However, we do not know if there is an analytic nodec q + (or SS) regular space.…”

“…Maximal topologies are not analytic. In fact, analytic topologies cannot be extremely disconnected or irresolvable, nevertheless there are nodec regular spaces with analytic topology [11,19].…”

“…Nodec regular spaces are not easy to construct, the most common examples are the maximal spaces. In [11] were constructed several examples of nodec regular spaces with analytic topology that are neither SS nor q + . However, it was left open whether there are nodec regular spaces satisfying other boolean combinations of the properties q + and SS.…”