# First countability, nets and sequences in the weak topology.

According to this book (see print below) the weak topology (as well as the weak* topology) is first countable. However the weak topology should be first countable only in the finite dimensional case. So, is the book wrong or am I missing something?

The book also says "with regard to convergence we may deal with sequences rather than nets". Is it wrong too?

• If you consider sets of the form $N(0;x_1^*,\ldots,x_n^*,1/m)$ for $n,m \in \mathbb N$ you get a neighborhood basis for $0$. – user42761 Jan 25 '17 at 15:15
• @Epsilon this is only countable for a finite dimensional space , because then we have a finite basis for the space of functionals. – Henno Brandsma Jan 25 '17 at 16:08
• – Henno Brandsma Jan 25 '17 at 16:13
• So we do need nets instead of sequences, for this case. – Henno Brandsma Jan 25 '17 at 16:15
• conclusion:the book is wrong – Henno Brandsma Jan 25 '17 at 16:19