Consider the set $X = \{1,2,3\}$, and the topologies $\mathcal{T}_1=\{\emptyset,\{2\},\{1,2\},\{2,3\},X\}$, $\mathcal{T}_2=\{\emptyset,\{1\},\{2,3\},X\}$. Determine whether $\mathcal{T}_1$ and $\mathcal{T}_2$ are compact.
By definition, a set $X$ is compact if every open cover of $X$ has a finite sub-collection that covers $X$. By this definition I see that both $\mathcal{T}_1,\mathcal{T}_2$ are open covers for $X$, since they are finite both have a finite sub-collection that covers $X$. Thus $X$ is compact with respect to both $\mathcal{T}_1$ and $\mathcal{T}_2$. Is this correct?