- #1
kazuyak
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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!
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!