TITLE: Example of different topologies with same convergent sequences
QUESTION [20 upvotes]: It's well known that for metric spaces the following is true
Let $ X $ be a space with two different metrics $ d_1,d_2$ such that the two topological spaces $ (X,d_1),(X,d_2) $ have the same convergent sequences. Then the two topologies are the same. 
Now looking at a space $ X $ with two topologies $ \tau_1,\tau_2 $ this is not true any more, i.e. if the topological spaces $ (X,\tau_1), (X,\tau_2)$ have the same convergent sequences the topologies may differ!
A classical example is $ l^1 $ due to Issai Schur.
So my questions are:

Is there a more topological/simpler example? 
I just know the "standard" functional analysis proof of Schur's lemma. On Wikipedia they refer to his article in "Journal für die reine und angewandte Mathematik, 151 (1921) pp. 79-111". I didn't work through the paper (and I won't) but just skimming through the paper, I don't see how we can deduce from the paper, that $ l^1$ has the Schur property. If there is someone who's familiar with the paper a short argument would be appreciated.

I'm not sure if I should place the second question here. If not, let me know and I will post a new one.
Thx and cheers
math

REPLY [3 votes]: Several nice examples have been given. I'll add a general construction, which can be used to produce more examples.
If $X$ is a space which is not sequential and $Y$ is the sequential coreflection of $X$, then $X$ and $Y$ have the same underlying set, they have the same convergent sequences and the topology of $Y$ is strictly finer than $X$.
The sequential coreflection of a topological space $X$ can be obtained in the following way:

Take all subspaces of $X$ of the form $\{x_n; n\in\mathbb N\}\cup\{x\}$ where $(x_n)$ is a sequence which converges to $x$ in $X$.
Make a direct sum of these spaces and then the quotient space obtained by identifying "all copies" of a given point $x$.

A picture would be better, but I hope with a little effort this construction can be understood from what I wrote. This is basically the construction given in Proposition 1.12 of S.P.Franklin: Spaces in which sequences suffice, Fund. Math. 57 (1965), 107-115.
The sequential coreflection can be equivalently described as the topology, in which closed sets are precisely the sequentially closed sets of the original topology. By sequentially closed we mean a set $A$ which for any convergent sequence of elements of $A$ contains all limits of this sequence.