Just a basic point-set topology question: clearly we can detect differences in topologies using convergent sequences, but is there an example of two distinct topologies on the same set which have the same convergent sequences? 



Is a topology determined by its convergent sequences?
In a metric (or metrizable) space, the topology is entirely determined by convergence of sequences. This does not hold in an arbitrary topological space, and Mariano has given the canonical counterexample. This is the beginning of more penetrating theories of convergence given by [nets](http://en.wikipedia.org/wiki/Net_(topology)) and/or [filters](http://en.wikipedia.org/wiki/Filter_(mathematics)). For information on this, see e.g.


[http://alpha.math.uga.edu/~pete/convergence.pdf](http://alpha.math.uga.edu/%7Epete/convergence.pdf)


In particular, Section 2 is devoted to the topic of sequences in topological spaces and gives some information on when sequences are "topologically sufficient".


In particular a topology *is* determined by specifying which nets converge to which points. This came up as a [previous MO question](https://mathoverflow.net/questions/19285/how-do-you-axiomatize-topology-via-nets). It is not covered in the notes above, but is well treated in Kelley's *General Topology*.


