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Answer

Can Nice Spaces be Broken into Two Equal Parts? Mikhail Matveev Asked in Can Nice Spaces be Broken into Two Equal Parts? by Stephen Watson

Assume \omega_n < c for all n \in \omega. For each n \in \omega, choose a subset A_n of the interval (1/(2n+1),1/(2n+2)) of the real line such that for \epsilon small enough, every \epsilon-neighbourhood of every point of this interval contains exactly \omega_n points of A_n. This is easy. Now put X = {0} \cup \cup {A_n:n \in \omega}. It is clear that the point 0 is unique in the sense of local cardinality, so the decomposition is impossible.