Failure of cancellation for connected sums of surfaces ($T^2\#{\mathbb{RP}}^2\cong {\#}^3{\mathbb{RP}}^2$)

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.