When identifying opposite sides of polygons we obtain a surface. For instance, the torus can be obtained by gluing together opposite sides of a square.
I saw a claim that if we identify 3 sides of a polygon then we do not obtain a surface. I don't see why this is true?
Could anyone explain why this is?
When you identify three sides of a polygon, then locally along the identified side you get a space that looks like three half-planes glued together along their boundary. You can visualize it as $\mathsf{Y}\times\mathbb{R}$ where $\mathsf{Y}$ is a space shaped like the letter Y (in other words, three line segments glued together at a point). That's not locally homeomorphic to $\mathbb{R}^2$ (though it takes some work to prove this rigorously), so it's not a surface.
This is in contrast to what happens when you identify two sides: then you get two half-planes glued together, which form one whole plane, so it is locally homeomorphic to $\mathbb{R}^2$.