A projective-plane plus Klein-bottle summand is the same surface as a projective-plane plus torus summand.Suggest a slogan
There is a homeomorphism \[ \RP^2 \# \KK \cong \RP^2 \# \TT^2 ,\] and both surfaces are homeomorphic to \(\RP^2\#\RP^2\#\RP^2\) [@Mun00].