Using Urysohn's Lemma, it can be shown that a connected normal space $X$ (with more than one point) is uncountable. But then how can it be that a connected normal space might just be a single point? Is this immediate from Urysohn's Lemma?
Connected normal space can just be a single point?
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general-topology
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1Urysohn makes no (nonvacuous) statement about one-point spaces. But the very definition of normal is trivially verified for one-point space. – 2012-11-11
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No, it’s immediate from the definitions of connectedness and normality. A one-point space is clearly not the union of two disjoint non-empty sets, open or otherwise, so it’s connected. A one-point space doesn’t contain two disjoint closed sets, so the defining condition of normality is vacuously satisfied.
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4@Sachin: Good. (By the way, what I wrote is basically just a longer version of what Asaf meant when he asked ‘Trivially?’.) – 2012-11-11