Generalization of $f: [0,1] \to [0, 1]$ has a fixed point? If $f: [0,1] \to [0,1]$ is a continuous map, there exists $x \in [0, 1]$ such that $f(x) = x$.  This can be proven by applying IVT to $f(x) - x$.  Does this fact generalize to continuous maps $f: X \to X$ where $X$ is a compact topological space?
 A: One generalisation is Brouwer's Fixed Point Theorem.
A: There are several generalizations of the result, but as Andre remarks, the sentence you stated is not true. One generalization of what you said is the Brouwer fixed-point theorem, which states that "every continuous function $f$ from a convex compact subset $K$ of a Euclidean space to $K$ itself has a fixed point." There is also the Lefschetz fixed-point theorem, which counts the fixed points of a continuous function via computations of the homology groups of a compact triangulable space. An interesting consequence of this theorem is that a generic map of such a space $X$ that is homotopic to the identity map will have $\chi(X)$ fixed points, where $\chi(X)$ is the Euler characteristic of $X$!
A: A connected counterexample is obtained by taking $X$ to be a circle or closed annulus in the plane and $f$ to be a rotation of $X$ about its centre through an angle $\theta$ that is not an integer multiple of $2\pi$.
A: Consider the case of the map $f: S^1\to S^1$ where every point is rotated by the same non-zero amount.
Edit: As Patrick correctly pointed out, the amount should be any value except for
integer multiples of $2$ $\pi$ (which includes 0)
A: Let $X=[0,1] \cup [2,3]$, and let $f$ take $[0,1]$ nicely to $[2,3]$, and vice-versa. 
