= Solution
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 example
$$
R(x,y)=(-x,-y),
$$
which 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.
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