Is a topology, generated by intersections of opens from two compact topologies, itself compact?

Question

Given two compact topological spaces $(X_1,\tau_1)$ and $(X_2,\tau_2)$, with nonempty $X_1\cap X_2=X_3$, let the topology $\tau_3$ on $X_3$ be generated by $\{O_1\cap O_2\mid O_1\in \tau_1, O_2\in \tau_2\}$. I suspect that $(X_3,\tau_3)$ need not be compact. If it is, could you give an argument? If it is not, can you give a counterexample and, perhaps, any conditions on $\tau_1,\tau_2$ under which $\tau_3$ is compact? ($X_3$ may be non-closed in one of the spaces.)

Answer 1

Let $X_1$ be the one-point compactification of $\mathbb{R}$: $X_1=\mathbb{R}\cup\{\infty\}$, $X_2$ be the extended real line $\bar{\mathbb R}=[-\infty,+\infty]$. Then $X_3$ is the usual $\mathbb{R}$ so not compact.

Of course, if $X_3$ is closed in both $X_1$ and $X_2$ then $X_3$ is compact (as $X_3\stackrel{\Delta}{\hookrightarrow}X_3\times X_3\hookrightarrow X_1\times X_2$)

Answer 2

As the other answerer has given a counterexample for general spaces, I want to mention when this is true.

Assume that $X_1$ and $X_2$ lie in some ambient Hausdorff space $X$ as subspaces. Certainly $X_3$ is closed as it is an intersection of closed sets. Moreover, a closed subspace of a compact space is compact, and so $X_3$ is compact.