New result today: there does exist a space first-countable
and sigma-ss, with no isolated points which is -not- accumulating.
First choose a maximal collection S of pairwise eventually distinct
sequences in N. Then take the usual "big grid" of (n,m) and (n,oo)
for n,m \in N. Add to it (as points) all the sequences in S.
if (s_n)_{n\in N} is a sequence in S, then the open sets containing
s are defined to be { (n,s_n) | n >= p } u { s } for all p \in N.
This space is first-countable and sigma-ss, and we can replace (n,m)
each with copies of R to prevent isolated points. Yet it is still
not accumulating, since nothing can accumulate to { (n,oo) | n \in N }
without containing some set lie { (n,t_n) | n \in N } which is cofinally
coincident with some s \in S, (do not be so would contradict maximality
of S) and so has s as an accumulation point also.
