Download Mathematics W4051x Topology

Mathematics W4051x
Topology
Assignment #6
Due October 21, 2011
1. (a) Let S and T be two topologies on the same set X with S ⊂ T . What does
compactness of X under one of these topologies imply about compactness under the
other? Give proofs or counterexamples.
(b) Show that if X is compact Hausdorff under both S and T , then S = T .
2. (a) Show that any topological space with the cofinite topology is compact.
(b) Let the cocountable topology on R be the topology under which U ⊂ R is open
if and only if either U = ∅ or its complement is countable. Show that R with the
cocountable topology is not compact.
3. A subset C ⊂ X of a metric space X is said to be bounded if there exists x0 ∈ X and
d0 ∈ R such that for all x ∈ C, d(x0 , x) ≤ d0 .
(a) Show that every compact subspace of a metric space is closed and bounded.
(b) Give an example of a metric space in which not every closed and bounded subset
is compact.
4. (20 pts) Let {An | n ∈ N} be a countable family of compact, connected subsets of a
Hausdorff space X such that An ⊃ An+1 for all n ∈ N. Let
A=
\
An .
n∈N
Prove that (a) A is nonempty if and only if each An is nonempty; (b) A is compact;
(c) A is connected.
Hint for (a): construct an open cover of A1 and pass to a finite subcover.
Hint for (c): if A = C ∪ D with C, D clopen, disjoint, and nonempty, choose disjoint
T
open sets U, V ⊂ X containing C and D respectively, and show that n∈N (An \ (U ∪V ))
is nonempty.