I read in an article for proof that the part $K$ of a topological space $(X, \tau)$ is compact. In this demonstration, the author proceeded to the following steps for the demonstration:
1- He showed that: $K$ is closed for $\tau $.
2- He showed that: On $K $ the topology $\tau $ coincides with a metrisable topology.
3- He showed that: $K$ is sequentially compact.
Therefore, $K$ is compact. I just think to show $(2)$ and $(3)$. I did not understand why the author showed that $K $ is closed?
An idea please.
My guess would be: $K$ is shown to be closed (probably it has to be anyway, when $X$ is Hausdorff e.g.), and the closedness of $K$ is used in the proof that $K$ is sequentially compact (e.g. because the convergent subsequence (of a sequence from $K$) is shown to exist in $X$ at first and then closedness of $K$ inside $X$ implies that the limit lies in $K$, which is necessary for the sequential compactness.
So you're right that 2.+3. would be enough in itself, but I think we might need 1. in the proof of 3. Hard to say for sure without looking at the proof itself, but at least my scenario would make sense.