A question on the size of the topology with respect to the sequential convergence.

Question

Let $\tau_1$ and $\tau_2$ be two topologies on a non-empty set $X.$ Let the sequential convergence with respect to one of the topologies be equivalent to the sequential convergence with respect to the other. Can we conclude that $\tau_1 = \tau_2\ $?

I don't think that it is correct. Our instructor hinted that we can construct a counter-example in the space $\ell^1(\mathbb N),$ the space of all summable sequences of complex numbers. But I can't figure it out. Could anyone give me some hint?

Thanks a bunch!

Answer 1

Here's an easy example. For $\tau_1$ take the discrete topology on $\mathbb{R}$. For $\tau_2$ let every point be isolated, except $0$ whose neighbourhoods will be the co-countable sets. In both topologies the convergent sequences are those that are constant eventually.

Answer 2

There are simpler examples that do not require functional analysis at all.

The cocountable topology on an uncountable set has this property that a sequence is convergent if and only if it is eventually constant (see this). The same property applies to the discrete topology which is not only different (in the sense of literal equality) but also not homeomorphic to the cocountable one.

For $\ell^1$ space you are dealing with the Shur's property. To see how $\ell^1$ has it read this: $\ell^1$ Schur property