Mathematics 4121A/9021A
Fall 2016
Problem Set 1
September 18, 2016.
1. (a) Let X be a three-element set. Define all possible topologies on X (up to permutation of
elements of X) and determine which pairs of them are comparable.
(b) How many distinct topologies are there (up to permutation) on a four-element set?
2. Is the family τ = {U ⊂ R : R \ U is infinite or empty} a topology on R?
T
S
3. If {τα }α is a collection of topologies on X, show that α τα is a topology on X. Is α τα a
topology on X?
4. Let {τα } be a collection of topologies on X. Show that there is a unique smallest topology τ 0
on X such that τα ⊂ τ 0 for all α, and a unique largest topology τ 00 on X such that τ 00 ⊂ τα for
all α.
5. (a) Give an example of a topological space (X, τ ) and a family of sets {Aι }ι in X such that
[
Aι 6=
[
ι
Aι .
ι
(b) A family of sets {Aι }ι in a topological space (X, τ ) is called locally finite, when every point
x ∈ X admits a neighbourhood U such that U intersects only finitely many of the Aι . Show
that if {Aι }ι is a locally finite family then
[
ι
Aι =
[
Aι .
ι
6. Let A = {1/n : n ∈ Z+ }. Find the set A0 of limit points of A in the finite-complement topology
on R. Do the same for the countable-complement topology.
7. Prove or give a counterexample:
(a) (A ∪ B)0 = A0 ∪ B 0
(b) (A ∩ B)0 = A0 ∩ B 0 .
8. Let β and β 0 be two bases for topology on a set X, and let τ and τ 0 be the topologies generated
by those bases, respectively. Show that β ⊂ β 0 implies τ ⊂ τ 0 but not the other way round.
9. Show that if β is a basis for topology on X, then the topology generated by β is the intersection
of all topologies on X containing β.
10. (a) Prove that in any T1 topological space (on a non-empty set) singletons are closed sets.
(b) Prove that a T1 topology on an infinite set does not admit a finite basis.
11. Show that every topological space (X, τ ) with topology τ induced by a metric on X admits a
basis β such that, for every x ∈ X, the basis of neighbourhoods β(x) is countable.
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