You say that a orientable surface (without border) is the sphere $S^2$ or connected sum of tori $T \# T \# \dots \# T$.
In case it is not orientable, is connected sum of projective planes $\mathbb{R}P^2 \# \mathbb{R}P^2 \# \dots \# \mathbb{R}P^2$.
So, what happens with the surface $T \# \mathbb{R}P^2$?
I know it is not orientable. But it is not expressed as unique connected sum of projective planes.
Please, help me.