Why this two spaces do not homeomorphic?

Question

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)

Already asked and answered in Math Overflow:
Are the rationals homeomorphic to any power of the rationals? — leo, Nov 09 '13 at 18:46
Possible duplicate: http://math.stackexchange.com/questions/476383/does-there-exist-a-continuous-bijection-mathbbq-to-mathbbq-times-mathbb/476387 — Prism, Nov 11 '13 at 21:13

score 10 · Accepted Answer · answered Nov 09 '13 at 18:24

10

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.

answered Nov 09 '13 at 18:24

Najib Idrissi

54,185

Why this two spaces do not homeomorphic?

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