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Rotation through any nonzero angle defines a continuous self-map of a circular annulus with no fixed point. Hence an annulus does not have the fixed-point property.

The Brouwer fixed-point theorem says that every continuous self-map of the closed unit disc has a fixed point.

A union of two disjoint closed discs does not have the fixed-point property. Define a map that is constant from the first component to a point in the second and constant from the second component to a point in the first. It is continuous because the components are disjoint and has no fixed point.

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.