Example of sequential space which is not Fréchet

Question

Definition:

$X$ is a sequential space if, whenever $A\subset X$ and $A$ is not closed, there is a sequence $\{a_n:n∈ω\}⊂A$ such that $a_n→y$ for some $y\in A^c$.

Is there any example to show that a sequential space need not be a Fréchet space?

Answer 1

Yes. The standard example is the Arens space, which is fully discussed in this post to Dan Ma’s Topology Blog. In fact, every sequential space that is not Fréchet contains a copy of the Arens space.