I need to know that the following statements if true or false:
Every homeomorphism of $S^2\rightarrow S^2$ has a fixed point.
Let $X$ be a complete metric space such that distance between any two point is less than $1$, Then $X$ is compact.
well, for 1 I see that it is false as $S^2$ is not convex so Brauer Fixed point Theorem can not be applied?
for 2 I thought that it will be compact, if not then my intuition says that it will be not sequentially compact or violates the definition of compactness?
Thank you for the help