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Since it is bounded and closed, it seems so. Massey's text says that

Theorem 5.1 [Algebraic Topology An Introduction] Any compact surface is homeomorphic to either a sphere or $n$-tori or the connected sum of $n$ projective planes.

But I cannot show annulus is homeomorphic to either one of them...

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Theorem 5.1 refers to surfaces without boundary. The annulus is a surface with boundary.

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