Let X be a topological space and A a subset of X . On X × {0, 1} deﬁne the

partition composed of the pairs {(a, 0), (a, 1)} for a ∈ A , and of the singletons {(x, i)} if x ∈ X \ A and i ∈ {0, 1}.

Let R be the equivalence relation deﬁned by this partition, let Y be the quotient space

[X × {0, 1}] /R and let p : X × {0, 1} → Y be the quotient map.

(1) Prove that there exists a continuous map f : Y → X such that f ◦ p(x, i) = x for every x ∈ X and i ∈ {0, 1} .

Prove that Y is Hausdorﬀ if and only if X is Hausdorﬀ and A is a closed subset of X .

(2) Consider the above construction for X = [0, 1] and A an arbitrary subset of [0, 1].

Prove that Y is compact. Prove that K = p(X × {0}) and L = p(X × {1}) are compact, and that K ∩ L is homeomorphic to A .

Any input is appreciated!