  A parametric version of Brouwer's Fixed Point Theorem, which is proven using
the fixed-point index, states that for every continuous mapping $f : (X \times
Y) \to Y$, where $X$ is nonempty, compact, and connected subset of a Hausdorff
topological space and $Y$ is a nonempty, convex, and compact subset of a
locally-convex topological vector space, the set of fixed points of $f$,
defined by $C_f := \{ (x,y) \in X \times Y \colon f(x,y)=y\}$, has a connected
component whose projection onto the first coordinate is $X$. In this note we
provide an elementary proof for this result, using its reduction to the case $X
= [0,1]$.
