A Certain Quotient Space

Question

Learning about the quotient maps and quotient spaces, the whole "gluing" process interested me. While experimenting, I struggled to visualize a space on the quotient space $[0,1] \times [0,1]/ \sim $ with the equivalence relation $(s,0) \sim (0, 1-s)$. where s $\in$ [0,1]. I tried to form 2 circles connecting the point $ (0,0) , (0,1)$ and $(1,0)$ together. Then, similar to the process of the space of Klein Bottle, I inverted one of the circles and connected it to the other one. Is my thought process correct? Is there an easier way of visualizing and does that space have a name?

Answer 1

You have constructed a Möbius strip, and the easiest way to see this is to cut up the square and re-glue it in a different way: