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Homework Statement
Show that a connected normal space having more than one point is uncountable.
The Attempt at a Solution
First of all, if X has more than one point, take the points a and b in X. Let U be an open set containing a. Take a basis element B around a contained in U. Then Cl(B) and X\U are disjoint closed sets containing a and b, respectively.
Since X is normal, by the Urysohn lemma there exists a continuous function f : X --> [0, 1] such that f(Cl(B)) = 0 and f(X\U) = 1.
Now, assume that X is countable, and let f(X) be the image set of f. f(X) is a countable subset of [0, 1], and since X is connected, by the intermediate value theorem, for any r between 0 and 1 there exists some point c of X such that f(c) = r. Let r be some point of [0, 1]\f(X). Then there must exist some c in X which maps to r. Since [0, 1] is uncountable, we arrive at a contradiction with the assumption that X is countable.
I hope this works, thanks in advance.