On some problems in general topology

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).