Let $X$ be a compact metric space with topology $\tau$ generated by the metric. Consider a new point $x_1\notin X$ and a non-isolated point $x_0\in X$. Set $\overline{X}=X\cup \{x_1\}$, equipped with the topology
\begin{equation} \overline{\tau}= \tau \cup \left\{W\cup \{x_1\}: x_0\in W, W\in\tau\right\}\cup \left\{(W\setminus \{x_0\})\cup \{x_1\}: x_0\in W, W\in \tau\right\}. \end{equation} I think that $(\overline{X}, \overline{\tau})$ is a compact space. Also It is not metrizable, because $\overline{X}$ is not a Hausdorff space.
Can we say that $\overline{X}$ is sequentially compact?
Please help me to know it.