M6140 Topology Exercises - 1st Week (2020)
1 Closed Sets
Exercise 1. Prove that an arbitrary intersection of closed sets is closed and that a finite union of
closed sets is closed.
Exercise 2. Show that a subset F of a topological space X is closed iff for each x ∈ F there exists
an open set U x such that U ∩ F = ∅.
Exercise 3. Let A, B be arbitrary subsets of a topological space X. Prove the following properties
of the closure.
(i) ∅ = ∅,
(ii) A ∪ B = A ∪ B,
(iii) A ⊆ A,
(iv) A ⊆ B implies A ⊆ B,
(v) A = A.
An operator on an arbitrary power set is called a closure operator if it satisfies the properties (iii),
(iv), (v), thus we now know that (−): P(X) → P(X) is a closure operator.
2 Topologies
Exercise 4. An Alexandrov topology is a topology in which an arbitrary intersection of open sets is
always open.
(a) Let X be a preordered set1. Prove that there is an Alexandrov topology on X such that the open
sets in X are precisely the lower sets2 in X.
(b) Let X be a topological space whose topology is Alexandrov. Prove that there is a preorder ≤ on
X defined by: x ≤ y iff y ∈ {x}.
(c) Prove that these two correspondences are inverse to each other.
Exercise 5. Consider the subsets of Z of the form S(a, b) := {an + b | n ∈ Z} of Z, where a is a
non-zero integer and b is an integer. Define a subset U of Z to be open iff for each b ∈ U there exists
a non-zero integer a such that S(a, b) ⊆ U.
1
A preorder is a reflexive and transitive relation.
2
A lower set in a preordered set is a subset such that if an element belongs to the subset, then all the lower elements
also belong to the subset.
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(a) Show that this defines a topology on Z. This topology is called the evenly spaced integer topology
or the Furstenberg topology.
(b) Show that each set S(a, b) is clopen.
(c) Show that each open set is either empty or infinite.
(d) Show that the complement of {−1, 1} is p prime S(p, 0).
(e) Conclude that there exist infinitely many primes.
Exercise 6. Suppose that P is a poset. Define a subset U of P to be open iff it is an upper set and
each directed set3 in P whose supremum belongs to U has a non-empty intersection with U.
(a) Show that this defines a topology on P. This topology is called the Scott topology.
(b) Show that a subset of P is closed iff it is a lower set that is closed under directed suprema in P.
(c) Show that a mapping P → Q between posets is continuous iff it preserves directed suprema.
Exercise 7. Let k be an algebraically closed field4 and let n be a positive integer. Define a subset F
of kn to be closed iff there exists an ideal I in the ring of polynomials over k of n variables such that
F = V (I), where V (I) := {x ∈ kn | ∀f ∈ I : f(x) = 0}. Show that in this way we obtain a topology
on kn. This topology is called the Zariski topology.
3 Continuous Maps
Exercise 8. Suppose that X and Y are topological spaces. Prove that if X is discrete, then each
mapping f : X → Y is continuous. Also prove that if Y is indiscrete, then each mapping f : X → Y
is continuous.
Exercise 9. Let f : X → Y be a continuous map. Show that the preimage of a closed set in Y is
closed in X.
Exercise 10. Show that a composition of continuous maps is continuous.
Exercise 11. Prove that Z and Q aren’t homeomorphic. Both topological spaces are viewed as
subspaces of R.
Exercise 12. A mapping f : X → Y between topological spaces is called continuous at a point x ∈ X
if for each neighbourhood N of the point f(x) its preimage f−1(N) is a neighbourhood of x. Show
that a mapping f : X → Y between topological spaces is continuous iff it is continuous at each point
of X.
3
A directed set in a poset is a non-empty subset such that each pair of elements of the subset has an upper bound in
the subset.
4
A field is called algebraically closed if each non-constant polynomial with coefficients from this field has a root in
this field.
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