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Texas A&M University Department of Mathematics Volodymyr Nekrashevych Fall 2015 Math 636 — Problem Set 7 Issued: 10.23 Due: 10.30 7.1. Find a subset of R homeomorphic to the one-point compactification of N. 7.2. Find a homeomorphism of the one-point compactification of 2N × N with 2N . 7.3. Show that the sequence 1, 2, 3, . . . has no converging subsequences in βN. Prove that βN has no converging sequences that are not eventually constant. 7.4. Let X be a completely regular spaces. Let A, B be subsets of X. Prove that their closures in βX are disjoint if and only if there exists a continuous function f : X −→ [0, 1] such that f |A = 0 and f |B = 1. 7.5. Let α be an ultra-filter on N. Order it by inclusion: A ≥ B iff A ⊂ B. Show that α with this order is a directed set. For every A ∈ α choose an element xα ∈ A. We get a net of points of N. Prove that this net converges to a point of βN.