= Failure of cancellation for connected sums of surfaces
{title2=$T^2\#\mathbb{RP}^2\cong\#^3\mathbb{RP}^2$}
The <torus> and <Klein bottle> are not homeomorphic because only the former is <orientable>. Adding a <projective plane> to either by <connected sum of surfaces> makes a <nonorientable surface> with <Euler characteristic> minus one. The <classification of compact surfaces> identifies both with three projective planes. Thus connected-sum cancellation can fail even for closed connected surfaces.
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