Fixed-point property

A topological space has the fixed-point property when every continuous self-map has a fixed point.

If $h:A\to B$ is a homeomorphism and $A$ has the fixed-point property, then a continuous $f:B\to B$ induces $h^{-1}fh:A\to A$. A fixed point of the conjugate map is carried by $h$ to a fixed point of $f$.

For continuous $f:[0,1]\to [0,1]$, the function $f(x)-x$ is nonnegative at zero and nonpositive at one, so the intermediate value theorem gives a fixed point.