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Let $h:A\to B$ be a homeomorphism, and let $f:B\to B$ be continuous. The conjugate mapis continuous. By hypothesis it has a fixed point $a\in A$. Thereforeand applying $h$ gives $f(h(a))=h(a)$. Hence every continuous self-map of $B$ has a fixed point. This is the homeomorphism invariance of the fixed-point property.

The closed unit disc has the fixed-point property by the Brouwer fixed-point theorem. The annulus admits the fixed-point-free rotation of an annulus, for examplewhich is continuous, maps the annulus to itself, and has no fixed point because the origin is not in the annulus. If the disc and annulus were homeomorphic, the homeomorphism invariance of the fixed-point property would give the annulus the fixed-point property, a contradiction. Therefore they are not homeomorphic.