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    $\begingroup$ I can't help expanding on my favourite briefly. The continuity of $\forall_X:(x\rightarrow\mathBB{S})\rightarrow\mathBB{S}$ for compact $X$ gives a way, due to Ulrich Berger, to exhaustively search certain infinite spaces in finite time. It's a beautiful meeting point of CS and topology. Some details here: math.andrej.com/2007/09/28/… $\endgroup$ Commented Mar 26, 2010 at 3:27
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    $\begingroup$ Thanks, I was under the impression that "rulers" or "imprecise measurements" requires a geometrical intuition, but from this post I see that it does not. I am completely happy with sigfpe's answer now. This was something I was trying to get at as well, except for this is much clearer. $\endgroup$ Commented Mar 26, 2010 at 4:06
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    $\begingroup$ The list of properties for discrete, Hausdorff, and compact beg for another: that the map $ \exists _ X \colon ( X \to \mathbb S ) \to \mathbb S $ be continuous. For topological spaces, this is always true, but it need not be in ASD. (A space with this property is called overt, a term from constructive locale theory.) $\endgroup$ Commented Jan 14, 2012 at 2:14
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    $\begingroup$ Oops, that URI up there should be paultaylor.eu/ASD since Taylor is a European individual and not an American university! $\endgroup$ Commented Jan 14, 2012 at 2:15
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    $\begingroup$ @QiaochuYuan This is embarrassingly ancient, almost five years old now, but I only just saw your question about internal logic of category of compact Hausdorff spaces. My immediate reaction is to say: that category is a pretopos (ncatlab.org/nlab/show/pretopos) since it is extensive and Barr exact; Barr exactness is by monadicity over $Set$. Being a pretopos means one can enact finitary first-order logic inside of it. $\endgroup$ Commented Mar 1, 2015 at 16:50