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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.