HOMEWORK 1
Obligatory exercises are 1,2 and 3.
Exercise 1. Show that SSk
= Sk+1
. Hint: Consider the map f : Sk
× I → Sk+1
,
where
f(x, t) = (( 1 − (2t − 1)2)x, 2t − 1)
Exercise 2. Let f : X → Y . Consider the mapping cylinder Mf .
(1) Prove that the inclusion ιX : X → Mf is a cofibration.
(2) Show that this gives a possibility to factor every map f : X → Y as
f = r ◦ ιX
where ιX is a cofibration and r: Mf → Y is a homotopy equivalence.
Exercise 3. Prove: If X is a Hausdorff space, then its diagonal
∆ = {(x, x) ∈ X × X}
is a closed subspace of X × X.
Exercise 4. Let X be a Hausdorff space and A ⊆ X be its retract. Prove that A
is closed. Hint: Use the previous exercise and the map X → X × X : x → (x, r(x))
where r: X → A is a retraction.
Exercise 5. The consequence of the previous exercise is: If X is Hausdorff and A → X
is a cofibration, then A is closed. Hint: X × I is Hausdorff.
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