Compactness is not box product-closed

This article gives the statement, and possibly proof, of a topological space property (i.e., compact space) not satisfying a topological space metaproperty (i.e., box product-closed property of topological spaces).
View all topological space metaproperty dissatisfactions | View all topological space metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for topological space properties
Get more facts about compact space|Get more facts about box product-closed property of topological spaces|

Statement

It is possible to have a collection $X_{i},i\in I$ of topological spaces, each of which is a compact space, but such that the product $\prod _{i\in I}X_{i}$, is not a compact space under the box topology.

Related facts

• Compactness is product-closed under the product topology. For finite products, this can be even more easily proved using the tube lemma