  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).
