Download Products and quotients via universal property

Products and quotients via universal property
This exercise consists of reinterpreting some of the results we’ve proved in class.
You may use results in the book without having to reprove them.
1. Let X and Y be topological spaces. Prove that X × Y has the following
universal property: if Z is a topological space and fX : Z → X and
fY : Z → Y are continuous functions, then there is a unique continuous
function h : Z → X × Y so that fX = pX ◦ h and fY = pY ◦ h where
pX : X × Y → X and pY : X × Y → Y are the projections.
2. Prove that X × Y is unique in the following sense. If W is another space,
with continuous functions qX : W → X and qY : W → Y , satisfying
the same universal property in 1), then there is a unique homeomorphism
g : W → X × Y such that qX = pX ◦ g and qY = pY ◦ g.
3. Let p : X → Y be a quotient map. Prove that Y has the following universal
property: if f : X → Z is a continuous function which is constant on each
p−1 (y) for y ∈ Y , then there is a unique continuous function h : Y → Z
so that f = h ◦ p.
4. State something similar to 2) for quotient spaces.
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