# Metrizable topologies

Are there two topologies on the same underlying set, exactly one of which is metrizable, which share the same convergent sequences with the same limits but are not the same?

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Let $E$ be an uncountable set, let $\tau_1=\mathcal P(E)$, and let $\tau_2=\{X\in\mathcal P(E):X=\emptyset\text{ or }|E\setminus X|\le\aleph_0\}$.
In either topology, a sequence $(x_1,x_2,x_3,\dots)$ converges to a point $x$ if and only if $x_n=x$ for all sufficiently large $n$. The discrete topology $\tau_1$ is metrizable, e.g., let $d(x,y)=1$ whenever $x\ne y$. The cocountable topology $\tau_2$ is not even Hausdorff, let alone metrizable.
P.S. If you want a countable example with some nontrivial convergent sequences, consider the following two topologies on $\mathbb N$:$$\tau_1=\{X\subseteq\mathbb N:1\notin X\}\cup\{X\subseteq\mathbb N:|\mathbb N\setminus X|\lt\aleph_0\};$$$$\tau_2=\{\emptyset\}\cup\{X\subseteq\mathbb N:|\mathbb N\setminus X|\lt\aleph_0\}\cup\{X\subseteq\mathbb N:1\notin X\text{ and }\sum_{n\in\mathbb N\setminus X}\frac1n\lt\infty\}.$$