Countable or uncountable numbers

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  The contradiction shows that the assumption that E is countable was wrong. We conclude that E is uncountable. Thank you for using Brainmass. This posting offers help to conclude that E is uncountable or countable.

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  Since this should be true for all x in X, and X is uncountable by our suumption, we conclude that U_n contains uncountably many open sets {x}. Contradiction. We conclude that X must be countable.