Sure.
Take [math]X=\mathbb{C}[/math], and [math]Y=\mathbb{C}-D[/math], where [math]D[/math] is some disk. (Open or closed, I don't care. Radius, I don't care.) Both have the usual topology.
Visibly [math]X[/math] is not homeomorphic to [math]Y[/math], since [math]X[/math] is simply...
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