Download Math 636 — Problem Set 7 Issued: 10.23 Due: 10.30

yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Birkhoff's representation theorem wikipedia, lookup

Basis (linear algebra) wikipedia, lookup

Fundamental theorem of algebra wikipedia, lookup

Covering space wikipedia, lookup

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
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.