I have a homework problem that asks us to find the basis $\mathbb{R}^2_{\rm discrete}$ with the lexicographic order topology. I actually have no idea, so any help is greatly appreciated. Thank you!
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1What class are you taking? – Brian Tung Mar 05 '21 at 02:13
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General Topology. – Ryan Sinclair Mar 05 '21 at 02:13
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Does this help? https://math.stackexchange.com/questions/789720/is-mathbbr2-lexicographic-cong-mathbbr-discrete-times-mathbbr – Brian Tung Mar 05 '21 at 02:15
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Kinda. The question is not the same. It's helpful tho. Thanks :) – Ryan Sinclair Mar 05 '21 at 02:17
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To be honest, I'm not sure exactly what you're asking. Do you want to find the basis of $\mathbb{R}^2_\text{discrete}$ with the lexicographic order topology? – Brian Tung Mar 05 '21 at 02:22
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Exactly what I'm trying to find. – Ryan Sinclair Mar 05 '21 at 02:26
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2I'm unfamiliar with the notation $\mathbb{R}^2_{\rm discrete}$. Care to explain? – Robert Shore Mar 05 '21 at 02:29
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1@RyanSinclair: The notation $\Bbb R^2_{\rm discrete}$ is rather odd. The lexicographic order topology on $\Bbb R^2$ is not the discrete topology on $\Bbb R^2$, nor is it $(\Bbb R_{\rm discrete})^2$ (which in any case is just $\Bbb R^2$ with the discrete topology). So what exactly do you mean by $\Bbb R^2_{\rm discrete}$? – Brian M. Scott Mar 05 '21 at 03:58