# Direct products of closures of subgroups

Let $H$ be subgroup of a topological group $G$. Suppose $H$ is the (internal) direct product of two of its subgroups $K_1$ and $K_2$. Does it follow that $\bar{H}$ is the direct product of $\bar{K_1}$ and $\bar{K_2}$?

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Well, depends on which direct product you mean, but if you mean only group product, you can find a very simple example.

Let $G=(\mathbb{R},+)$. Let $K_1=\mathbb{Z}$ and $K_2=\alpha\mathbb{Z}$ where $\alpha$ is irrational. Then $H = \{u+v\alpha: u,v\in\mathbb{Z}\} \cong K_1\oplus K_2$. But $H$ is dense in $\mathbb{R}$, so $\bar{H}=\mathbb{R}$, but $K_1$ and $K_2$ are closed in $\mathbb{R}$.

In this case $H$ is not homeomorphic to $K_1\times K_2$ as a topological space. I'm not sure what you can say if you require that the group isomorphism between $H$ and $K_1\oplus K_2$ is also a homeomorphism.

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Yes this is what I meant. Thanks for the example. –  Mustafa Gokhan Benli Sep 30 '11 at 15:58