Topology Problems

Question 1
Let I be the closed unit interval, [0, 1] := {t ∈ R | 0 ≤ t ≤ 1}, endowed with its Euclidean topology. A path in the topological space (X, T), from a ∈ X to b ∈ X is a continuous function
γ: I --> X with γ(0) = a and γ(1) = b.
The subset A of the topological space (X, T) is path-connected if and only if for all a, b ∈ A, there is a path joining a to b in A.
(i) Prove that every path-connected set is connected.
(ii) Determine whether the subset
{(x, sin(1/x)) | 0 < x <1 } U {(0, y) | -1 ≤ y ≤ 1} or R² is
(a) connected,
(b) path-connected.

Question 2.
Prove that the product of finitely many topological spaces is connected if and only if each factor is connected.

Question 3.
Let (X, ρ) and (Y, σ) be metric spaces.
Prove that if f: X --> Y is uniformly continuous, then it maps Cauchy sequences to Cauchy sequences.

Question 4.
Let (X, ρ) and (Y, σ) be metric spaces.
Prove that if X is compact, every continuous function f: X --> Y is uniformly continuous.

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