Algebraic Topology problem help! A mapping from the n-sphere to some set!

Question

So let $f:S^{m}\to X$ be a continuous mapping. How can I prove that a) $f$ is homotopic with the constant mapping (i.e. the point I guess) and b) that $f$ can be augmented to a new mapping $f': D^{m+1}\to X$ (i.e. from the unit disk in $m+1$ dimensions to $X$) are equivalent?

Answer 1

A hint to get you going: Let $H:S^m \times I \to X$ be a homotopy from $f$ to a constant map. Then $H$ is constant when restricted to $S^m \times \{1\}$, so it factors through the quotient space $S^m\times I / S^m \times \{1\}$ (i.e. it induces a map from this space to $X$). But that quotient space is exactly $D^{m+1}$! Can you see why?