On some problems in general topology

TL;DR

This paper demonstrates the consistency of certain topological space properties under set-theoretic assumptions, showing how forcing can produce Lindelöf spaces with specific characteristics in models where CH holds.

Contribution

It proves the consistency of the existence of Lindelöf regular spaces with particular bases and pseudo-character properties under CH and forcing, advancing understanding in set-theoretic topology.

Findings

Existence of Lindelöf regular spaces with clopen bases and singleton intersections under CH.

Consistency results involving forcing notions and topological space properties.

Non-existence of such spaces under certain cardinal assumptions, starting from a weakly compact cardinal.

Abstract

We prove that Arhangelskii's problem has a consistent positive answer: if V\models CH, then for some aleph_1-complete aleph_2-c.c. forcing notion P of cardinality aleph_2 we have that P forces ``CH and there is a Lindelof regular topological space of size aleph_2 with clopen basis with every point of pseudo-character aleph_0 (i.e. each singleton is the intersection of countably many open sets)''. Also, we prove the consistency of: CH+ 2^{aleph_1} > \aleph_2 + "there is no space as above with aleph_2 points" (starting with a weakly compact cardinal).