M6140 Topology Exercises - 6th Week (2020)
1 Homotopy
Definition 1. For topological spaces X, Y we denote by [X, Y ] the set of homotopy classes of continuous
maps X → Y .
Exercise 1. Prove that [∗, Rn \ {0}], where n > 1, is a singleton.
Exercise 2. Show that [∗, R \ {0}] is a two-element set.
Exercise 3. Show that [X, I] is a singleton for each topological space X.
Exercise 4. Prove that if Y is path-connected, then [I, Y ] is a singleton.
Definition 2. A subset A of Rn is called star-shaped if there exists a point a ∈ A such that for each
x ∈ A the line segment joining a with x is contained in A.
Exercise 5. Prove that [X, A] is a singleton for each topological space X and each star-shaped subset
A of Rn.
Exercise 6. Suppose that A is a star-shaped subset of Rn and Y is a path-connected topological
space. Show that [A, Y ] is a singleton.
Exercise 7. Show that each non-surjective continuous map X → Sn, where X is a topological space,
is null-homotopic, i.e. homotopic to a constant.
Exercise 8. Let X be a topological space and let f, g be continuous maps X → C \ {0} such that for
all x ∈ X: |f(x) − g(x)| < |f(x)|. Prove that f and g are homotopic.
Exercise 9. Let X be a topological space and let f, g be continuous maps X → Sn such that for all
x ∈ X: |f(x) − g(x)| < 2. Show that f and g are homotopic.
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