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Advanced Analysis Spring 2006 Exercises for Feb 1st: E1: Let S be a topological space. We say that S is connected if it cannot be decomposed as a union of two nonempty disjoint open sets. A subset of S is called clopen if it is both open and closed. Show that S is connected if and only if S and ∅ are the only clopen sets in S. E2: Let S be a topological space. If A ⊆ S and x ∈ S then x ∈ A if and only if A ∩ N 6= ∅ for every neighborhood N of x. 1