see image Prove that if X is connected regular Ti-space, then X is uncountable T

Question

see image

Prove that if X is connected regular Ti-space, then X is uncountable This says that there are essentially no connected spaces. A space is called connected if it is not disconnected. (x, ) X is disconnected iff non-empty disjoint open sets H,K of X with X = H K. A space is called completely regular iff for any closed A X and any point XA there is a continuous function f:X rightarrow [0,1] such that f(x)=0 and A f-1(1). (X, ) is called a Ti-space iff for two distinct points X,Y X there exits open sets u,v where x u, x v and y v and y u.

Explanation / Answer

So let a and b be two different points in the connected completely regular T1 space X. Then
we can separate a and {b} by a continuous function f: X--->R such that f(a) = 0 and f[{b}] = 1.
So f[X} is connected subset of R (because f is continuous and X is connected). Also f[X] contains
more than one point. But the only connected subsets of the reals with more than one point are the
non-trivial intervals. Every non-trivial intervals is uncountable, in fact it has the same
cardinality as R. So we have a function from X onto the uncountable set f[X]. Thus x must be
uncountable, in fact it must have at least the cardinality of R.

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