Acyclicity is product-closed

For two spaces
Suppose $$X_1$$ and $$X_2$$ are topological spaces that are both acyclic spaces. Then, the product space $$X_1 \times X_2$$, endowed with the product topology, is also an fact about::acyclic space.

For finitely many spaces
Suppose $$X_1, X_2, \dots, X_n$$ are topological spaces that are all acyclic spaces. Then, the product space $$X_1 \times X_2 \times \dots \times X_n$$, endowed with the product topology, is also an acyclic space.

For an arbitrary number of spaces
Suppose $$X_i, i \in I$$, are topological spaces that are all acyclic spaces. Then, the product space $$\prod_{i \in I} X_i$$, endowed with the product topology, is also an acyclic space.

Related facts

 * Rational acyclicity is product-closed
 * Contractibility is product-closed
 * Weak contractibility is product-closed
 * Connectedness is product-closed
 * Path-connectedness is product-closed
 * Simple connectedness is product-closed