Mock paper

5CCM226A Metric spaces and topology

Summer 2024

SECTION A

A 1. Let (X, d) be a metric space.

(a) Define a topology on X. [3 marks]

(b) Define the topology T generated by d on X, and check that it satisfies the axioms of topology you gave in (a). [6 marks]

(c) For each of the following statements determine whether it is true or false. Justify your answers with a proof or a counterexample. A brief argument should normally be sufficient.

i. If ρ(x, y) = 2/1d(x, y), then ρ is a metric on X generating the same topology T .

ii. If ρ is another metric on X generating the same topology T , then for every x, y ∈ X, d(x, y) = ρ(x, y).

iii. If ρ is another metric on X generating the same topology T , then ρ is equivalent to d.

iv. If ρ is another metric on X that is equivalent to d, then ρ generates a topology that is strictly finer than T .   [16 marks]

A 2. Let (X, d) be a metric space.

(a) Define the diameter of a set A ⊆ X, and state Cantor’s lemma, including all its assumptions.   [4 marks]

(b) Prove that if T : X → X is a contraction, and x0, x1 are fixed points of T, then x0 = x1.   [5 marks]

(c) For each of the following statements determine whether it is true or false. Justify your answers with a proof or a counterexample. A brief argument should normally be sufficient.

i. The map T : R → R defined by T(x) = ax + b for some a, b ∈ R, is a contraction if and only if |a| < 1.

ii. If T : R → R is a contraction, then necessarily T is of the form. T(x) = ax + b, for some a, b ∈ R.

iii. If T : R → R is a contraction, then the map S : R → R defined by S(x) = T(x) + 2024 is also a contraction.

iv. If T : R → R is a contraction, then the map S : R → R defined by S(x) = 2T(x) is also a contraction.   [16 marks]

SECTION B

B 3. (a) Define what it means for a topological space (X, T ) to be Hausdorff, and prove that if X is Hausdorff, then every finite set A ⊆ X is closed. [5 marks]

(b) i. Define the finite complement topology on a set X. (You are not expected to prove that it is a topology.)

ii. Let Rf = (R, Tf ) be the set of real numbers, equipped with the finite complement topology. Check whether every finite set A ⊆ R is closed in Rf , and whether Rf is Hausdorff. Hence determine whether the converse to (a) is true, i.e. whether there exist spaces X, so that every finite set A ⊆ X is closed, but X is not Hausdorff.

iii. Determine which of the sequences xn = (−1)n and yn = n/1 are convergent in Rf , and for those convergent sequences find all the limits.   [10 marks]

(c) i. Define the “countable complement topology” on a set X. (You are not expected to prove that it is a topology.)

ii. Is R equipped with the countable complement topology Hausdorff?

iii. Is R equipped with the countable complement topology metrizable?

iv. Is R equipped with the countable complement topology connected?

v. Is R equipped with the countable complement topology compact?       [10 marks]

B 4. (a) Determine whether the map T(x) = x 2 a contraction on the following metric spaces:

i. R.

ii. (−1/2, 1/2) ⊆ R

iii. [−1/4, 1/4] ⊆ R.

[5 marks]

(b) i. Define the “lower limit topology” on R, and denote the resulting topological space RL.

ii. Does the standard metric on R generate RL?

iii. Is the sequence xn = (− n 1)n convergent in RL? Explain, and find all of its limits if it is indeed convergent.

iv. Is the function f(x) = x 2 continuous as a function f : RL → R?     [10 marks]

(c) i. Define a totally bounded metric space.

ii. Let X = Q ∩ [0, 1] ⊆ R. Determine whether X is totally bounded, and whether X is complete.

iii. Thus deduce whether X is sequentially compact, and whether it is compact as a topological space.

iv. Determine whether X = Q ∩ [0, 1] is connected, and whether it is path-connected.    [10 marks]