Acyclicity is product-closed

This article gives the statement, and possibly proof, of a topological space property (i.e., acyclic space) satisfying a topological space metaproperty (i.e., product-closed property of topological spaces)
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Statement

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 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 $X_{1} \times X_{2} \times \dots X_{n}$ , 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.

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