TITLE: Why this two spaces do not homeomorphic?
QUESTION [7 upvotes]: Consider $\Bbb Q$ with subspace topology and $\Bbb Q\times \Bbb Q$ with product topology. Why this two spaces are not homeomorphic?($\Bbb Q$ is the rational numbers)

REPLY [10 votes]: It is a theorem of Sierpinski that every countable metric space without isolated points is homeomorphic to $\mathbb{Q}$. See eg here. $\mathbb{Q}^2$ is a countable metric space without isolated points. Therefore these two spaces are homeomorphic.