Let $V = \{f:[0,1]\to \mathbb{R}: f \text{ is continuous}\}$ and consider the metric that is defined for $f,g\in V$ by $$d(f,g) = \max\{|f(t)-g(t)|: t\in [0,1]\}.$$ We set $E = \{\{f,g\}: f,g \in V\text{ and } d(f,g) = 1\}$. Setting $G:=(V,E)$ it is easy to see that $G$ has a countable clique, but do we also have $\chi(G) = \aleph_0$?
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asked Mar 2, 2015 at 8:19
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$\begingroup$ It is a separable space and so may be covered by countably many sets of diameter less then 1. $\endgroup$– Fedor PetrovCommented Mar 2, 2015 at 8:38
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$\begingroup$ Oh - thanks. Can you put this as an answer so we can close the question? $\endgroup$– Dominic van der ZypenCommented Mar 2, 2015 at 8:52
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It is a separable space and so may be covered by countably many sets of diameter less then 1.
answered Mar 2, 2015 at 9:01