Example 2

In a topological space $X$, let $(x_n)_{n \in \mathbf{N}}$ be an infinite sequence of points which converges to a point $a$; then the set $A$ consisting of the points $x_n$ ($n \in \mathbf{N}$) and $a$ is quasi-compact. For if $(U_i)$ is a covering of $A$ by open sets of $X$, then $a \in U_x$ for some index $x$. $U_x$ is a neighbourhood of $a$ and therefore there is only a finite number of indices $n_k$ such that $x_{n_k} \notin U_x$. For each index $k$ let $i_k$ be an index such that $x_{n_k} \in U_{i_k}$; then $U_x$ and the $U_{i_k}$ form a finite open covering of $A$.