From $[0,1]\times [0,1]$ construct the Klein bottle

Question

The argument by Wikipedia is enough or complete for the solution of the exercise?

Exercise: From $[0,1]\times [0,1]$ construct the topological space known as Klein bottle.

Wikipedia says

...More formally, the Klein bottle is the quotient space described as the square $[0,1] × [0,1]$ with sides identified by the relations $(0, y) \sim (1, y)$ for $0 ≤ y ≤ 1$ and $(x, 0)\sim (1 − x, 1)$ for $0 ≤ x ≤ 1.$

Answer 1

Well, that answer looks just fine as I see it. But if you want more detail, you can extend on the idea given by Wikipedia. I will call $[0,1]=I$ for short.

We start with the square $[0,1]\times [0,1]=I^2$. We define an equivalence relation $\sim$ on it, the one generated by $(0,y)∼(1,y)$ for $0≤y≤1$ and $(x,0)∼(1−x,1)$ for $0≤x≤1$ (verify it is an equivalence relation).

Let $g:I^2\rightarrow I^2/\sim$ be the canonical quotient map, and give $I^2/\sim$ the quotient topology with respect to $g$. Geometrically you are identifying or "gluing" the sides of the square as follows:

You then have a completely defined topological space (the Klein bottle) built from the square $I^2$.

Answer 2

You take the unit square and identify opposite sides to get a cylinder. Then go to identify the two circles at the opposite ends of the cylinder, like you were going to make a donut (torus), but, twist one of the circles $180^0$ before identifying...