Problems-contractible spaces

Question

Problem:

Contractible spaces

Let X be a contractible space:

a) Show that X is path connected

b) Show that any two continuous maps ƒ,g: Y→X  where Y is any topological space, are homotopic.

c) Let x₁∈ X and C_x1 : X→X be the map defined by C_x1(x) = x₁. Show that id_x and C_x1 are homotopic.