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Math Help - Locally Path Connected

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    Locally Path Connected

    Are the rational numbers Q, with the relative topology as a subset of the real numbers, locally compact ? why?
    Last edited by lttlbbygurl; May 25th 2010 at 11:25 PM.
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    Quote Originally Posted by lttlbbygurl View Post
    Are the rational numbers Q, with the relative topology as a subset of the real numbers, locally compact ? why?
    You wrote locally path connected, when I assume you meant compact like you said in the actualy body.

    The answer is no. Let U be any neighborhood of 0 in \mathbb{Q}. Then, \text{cl}_\mathbb{Q}\text{ }U=\mathbb{Q}\cap\text{cl}_\mathbb{R}\text{ }U. Also, remember then that if X,Y,Z are top. spaces and X is a superspace of Y then Z will be a compact subspace of Y if and only if it's a compact subspace of X and thus for \text{cl}_\mathbb{Q}\text{ }U to be compact in \mathbb{Q} we need that \mathbb{Q}\cap\text{cl}_\mathbb{R}\text{ }U is compact in \mathbb{R}. But, evidently this is not compact in \mathbb{R}, why?
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