$$\text{Cut a Möbius band from its center line, and then what do we get?}$$
Someone may find it's not easy to imagine without a paper in hand. However, if we cut a square paper from center line at first, and glue it as a Möbius band, we will easily see that it is a cylinder.
We can also add the number of cut lines, like three (three points) and four (four points). Now what will Möbius band be like?
It is difficult to answer from the first method, but easy from the second. In the three case, we will find there are two disconnected bands, one is a Möbius band, other is a cylinder.
Actually, the two methods can be simplified as below
First paste a paper into a Möbius band, and then cut it.
First cut a paper into pieces, and then paste it.
So my question is
Are the two methods or viewpoints same under the homoemorphism?
I want to find a rigorous proof. Any advice is helpful. Thank you.
The point is to define rigorously gluing and cutting. For gluing, one uses the notion of quotient topology $X/\sim$, for some relation $\sim$. For cutting, one can just remove a subset, for example if $X$ is the unit square $X=\{(x,y)|0\leq x,y\leq1\}$, then after cutting we can get $Y=X\setminus\{y=1/2\}$.
Examine the question using the above paragraph. You can either take quotient first and then remove a subset, or first remove a subset and then take quotient. Now it is easy to verify that with the correct notations, you get exactly the same space.