Show that if $n>1$, every continuous map $f:\mathbb{S}^n\to \mathbb{S}^1$ is nulhomotopic, and show that every continuous map $f: P^2\to \mathbb{S}^1$ is nulhomotopic.
I have already demonstrated the following result and I would like to know if I can use that here:
I know that $\pi_1(X)$ is finite if $X=\mathbb{S}^n$ or $X=P^2$, I also know that $\mathbb{S}^n$ is arc-connected, but are the other things in the hypothesis fulfilled in order to apply this result?
First, $\mathbb S^n$ is locally path connected, for any neighborhood of a point of $\mathbb S^n$ contains an $\varepsilon$-ball of the point, which is homeomorphic to $\mathbb R^n$.
Second, $P^2$ is path connected, for it is the image of $\mathbb S^2$ under a continuous (quotient) map.
Third, $P^2$ is locally path connected. This is because any neighborhood of a point contains a neighborhood on which the preimage of the aforementioned quotient map is homeomorphic to two copies of $\mathbb R^2$, hence is homeomorphic to $\mathbb R^2$.