 Let \( \displaystyle f : X \longrightarrow Y \) be a continuous map between two topological spaces. Prove that \( \displaystyle f \) induces a mapping \( \displaystyle f_{*} : \pi_{1}(X,x_{0}) \longrightarrow \pi_{1}(Y,f(x_{0})) \) of the fundamental groups, which is a group homomorphism. Additionally, show that if \( \displaystyle f \) is a homeomorphism, then \( \displaystyle f_{*} \) is an isomorphism.
 Prove that the Riemann sphere is arcwise connected, and subsequently prove that it is simply connected.
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