A sphere with one handle, one cross cap and two holes is shown below.
To detach the handle leaves two holes and the crosscap one hole. On a sphere with n holes, m handles and c crosscaps, n+2m+c holes are left by detaching all the handles and crosscaps. When the handles are detached they become cylinders with Euler characteristic zero.
Hence
A surface expressed as a sphere with n holes, m handles and c crosscaps is said to be in standard form. Some surfaces are represented in standard form in the following table.
Surface 
holes 
handles 
Crosscaps 

Sphere 
0 
0 
0 
2 
Disc 
1 
0 
0 
1 
Cylinder 
2 
0 
0 
0 
Real Projective Plane 
0 
0 
1 
1 
Mobius Band 
1 
0 
1 
0 
Torus 
0 
1 
0 
0 
Two Hole Torus 
0 
2 
0 
2 
n Hole Torus 
0 
n 
0 
22n 