Let $X$ be a compact connected Hausdorff topological space.

We say $X$ is a cohomologicaly minimal space, briefly CM space, if $X$ satisfies the following property:

"For every proper subset $A\subset X$ with the inclusion map $i:A \to X$, $i^*$ is NOT a ring isomorphism between $H^*(X,\mathbb{Z})$ and $H^*(A,\mathbb{Z})$"

In the other word, $A$ is the only subset of $A$ whose inclusion gives a ring isomorphism cohomology.

Examples; All closed manifolds (see A closed manifold with a subset with the same ring cohomology )

Non Examples: Closed disc, figure 8,...etc.

My question

Is the product of two CM spaces, a CM space?