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MATH3032: Topology and Analysis – 2023

ASSIGNMENT 1. Due by 5pm, Friday 25th of August 2023. The number of marks available for each question is shown below. The total number of marks available is 40 (worth 14 % for the final assessment). Please send a pdf file of your assignment (with your name and student number) to: [email protected] .

Remark. Recall that the metric topology Td on R n determined by the standard metric d is called the standard topology on R n .

1. For each of the statements below determine whether it is true or false, providing reasons for your answer.

(a) For the real line R with the standard topology and its subset Q of all rational numbers we have Cl(Q) = R. [2 marks]

(b) For the real line R with the standard topology and its subset Q of all rational numbers we have ∂(Q) = R. [2 marks]

Note: You can use the fact that every non-empty open interval in R contains both rational and irrational numbers.

2. Let

T = {(a,∞) : a ∈ [−∞,∞]}.

(Note: when a = −∞ we have (a,∞) = R, while if a = ∞, then (a,∞) = ∅.)

(a) Show that T is a topology on R. [5 marks]

(b) Carefully explain whether T is Hausdorff or not. [2 marks]

3. Let X be a topological space and let K1, K2, . . . , Km be compact subsets of X. Show that

K = K1 ∪ K2 ∪ . . . ∪ Km

is compact, too. [5 marks]

4. Let X be a topological space. Prove that

Int(A ∩ B) = Int(A) ∩ Int(B)

for all subsets A and B of X. Recall that Int(A) denotes the interior of the set A, i.e. the largest open subset of X contained in A. [8 marks]

5. Let S n be the unit sphere with centre at 0 in R n+1 and let N = (0, . . . , 0, 1) ∈ R n+1 .

The stereographic projection

p : S n \ {N} −→ R n = R n × {0} ⊂ R n+1

is such that for each x ∈ S n \ {N}, the point p(x) is the intersection of the line determined by N and x with the hyperplane xn+1 = 0. Find an explicit formula for p(x) by means of the coordinates of x, and also an explicit formula for the inverse function p −1 , and prove that p is a homeomorphism. (This will show that S n \ {N} and R n are homeomorphic.) [8 marks]

6. Let (X, T ) be a Hausdorff topological space and let

K1 ⊇ K2 ⊇ . . . ⊇ Kn ⊇ . . .

be an infinite sequence of non-empty compact subsets of X. Show that

∞

\ n=1 Kn = ∅.

i.e. there exists a point x ∈ X such that x ∈ Kn for all n ≥ 1. [8 marks]

Hint: Assume ∩ ∞ n=1Kn = ∅ and consider the sets Un = X \ Kn. Then use the compactness of K1.