TITLE: connected sum of torus with projective plane
QUESTION [19 upvotes]: I would like to understand how to prove that the connected sum $\mathbb{R}P^2 \# T^2$ of the projective plane with a torus is homeomoprhic to $\mathbb{R}P^2 \# \mathbb{R}P^2 \# \mathbb{R}P^2$. 
I got as far as showing that it must be equivalent to a connected sum of projective planes, how can I argue though that I need precisely three projective planes ?
Thanks for your help! 
(P.S. not a homework exercise, this is for me to understand the classification of surfaces).

REPLY [24 votes]: The following picture will help you to understand this problem intuitively. (If you can understand why $\mathbb{R}\mathrm{P}^2 \# \mathbb{R}\mathrm{P}^2 = K$, where $K$ stands for the Klein bottle.)
The figure in the upper right corner is $K \setminus \text{disk} $. It may take some time to think why it looks like this.

Source of the picture: link