Let $\mathscr T\_1$ and $\mathscr T\_2$ be two topologies on space $X$. Assume that $(X,\mathscr T\_1)$ is metrizable, and any sequence in $X$ that converges in one of the two topologies must also converge in the other topologies, i.e., $$(\,\forall\{x\_n\}\_{n\in\mathbb N}\subset X\,)\Big(\big(x\_n\xrightarrow[]{\mathscr T\_1} x\big) \Leftrightarrow \big(x\_n\xrightarrow[]{\mathscr T\_2} x\big)\Big).$$

The question is, whether $\mathscr T\_1$ and $\mathscr T\_2$ are the same topology on $X$?

The answer should be affirmative, since this argument is used in many analysis books without hesitation, such as [the proof of metrizable of locally convex spaces in Conway's book of functional analysis](https://books.google.com/books?id=ccEGCAAAQBAJ&pg=PA108&dq=A+course+in+functional+analysis+%222.1.+Proposition.+Let+%22&hl=zh-CN&sa=X&ved=0ahUKEwjmoNDcx-LWAhUqwlQKHXpXAIMQ6AEIJzAA#v=onepage&q=A%20course%20in%20functional%20analysis%20%222.1.%20Proposition.%20Let%20%22&f=false). But I don't know how to prove.

Any comments or hints will be appreciated!

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**Edit:** Very sorry! I'll explain here the details I thought about the link of Conway's book. The motivation is from the proof of Proposition IV.2.1 in that book as linked. 

To prove the necessity part of the proposition, that is, 

> 
> If a local compact space $X$ is metrizable, then its topology is determined by a countable family of seminorms.
> 
> 
> 

the author construct a countable family of seminorms $\{p\_n\}$, and then show that $\{p\_n\}$ determine the same sequential convergence as a given metric $\rho$ of $X$, that is, 
$$(\,\forall\{x\_n\}\_{n\in\mathbb N}\subset X\,)\Big(\big(x\_n\xrightarrow[]{\{p\_n\}} x\big) \Leftrightarrow \big(x\_n\xrightarrow[]{\rho} x\big)\Big).$$
But how to assert from the preceding statement that $\{p\_n\}$ determine the same topology as the original one?