Caution
We learnt that, for metric spaces, sequential convergence was adequate to
describe the topology of such spaces (in the sense that the basic primitives
of `open set', `neighbourhood', `closure' etc. could be fully characterised
in terms of sequential convergence). However, for general topological spaces,
sequential convergence fails. We illustrate:
(i) Limits are not always unique. For example, in 
,
each sequence 
converges to every 
.
(ii) In R with the cocountable topology 
, [0,1] is not
closed and so 
is not open -- yet if 
where 
, then Assignment 1 shows that 
for
all sufficiently large n.
Further, 
, yet no sequence in [0,1] can
approach 2. So another characterisation fails to carry over from metric space
theory.
Finally, every -convergent sequence of points in [0,1] must have
its limit in [0,1] -- but [0,1] is not closed (in )!
Hence, to discuss topological convergence thoroughly, we need to develop a new basic set-theoretic tool which generalises the notion of sequence. It is called a net -- we shall return to this later.
The investigations above show that and 
are
examples of non-metrizable spaces. However, the discrete space 
is metrizable, being induced by the discrete metric
