Same homorphism group fundamental

Question

Prove that two continuous mappings φ, ψ:X→Y, with φ(x0)=ψ(x0) for some point x0∈X, induce the same homomorphism from π(X,x0) to π(Y,φ(x0)) if φ and ψ are homotopic relative to x0.

Answer 1

A couple of things to think about that should lead you to an answer:

1. What is an element of a fundamental group?
2. Given a continuous function $f:X\to Y$, how does the induced homomorphism $\pi_1(X,x_0)\to \pi_1(Y,f(x_0))$ work?
3. Given two loops in $Y$ based at $\psi(x_0)$, what does it mean that the corresponding elements in $\pi_1(Y, \psi(x_0))$ are equal?
4. What was the relation between $\psi$ and $\varphi$ again? What does that mean, specifically?