The following is from Munkres Topology: 

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> A space X is completely regular if and only if it is homeomorphic to a subspace of $[0,1]$$^J$ for some $J$. 
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Note: $[0,1]$$^J$ is given the product topology.

I realize that $[0,1]$$^J$ is a completely regular space and any subspace of a completely regular space is also completely regular. 

However, why can we conclude for a space X that is homeomorphic to $[0,1]$$^J$ that it must also be completely regular?

I don't find the answer that homeomorphisms "preserve topological properties" convincing enough and am looking for something more specific. 

Thanks!