About relation between continuity and sequentially continuity

Question

Its is known every continuous function between topological spaces is sequentially continuous. Also sequentially continuity implies continuity if the domain space is first countable. I am interested in example of the function between topological spaces (where domain space is not first countable, as I say) that is sequentially continuous but not continuous.

Answer 1

On $\mathbb R$, consider the topology $\tau$ for which a set $A$ is open if and only if $A=\emptyset$ or $A^\complement$ is countable. Let $\tau_d$ be the discrete topology. For both of them, a sequence $(x_n)_{n\in\mathbb N}$ converges to some $x$ if and only of $x_n=x$ if $n$ is large enough. So, the identity from $(\mathbb R,\tau)$ into $(\mathbb R,\tau_d)$ is sequentially continuous. But it is not continuous, of course. For instance, $(0,1)$ is an open subset of $(\mathbb R,\tau_d)$, but not of $(\mathbb R,\tau)$.