Does cutting and pasting preserve continuity in the quotient space.

Question

I have a hexagon with interior with the opposite sides being identified. In order find the identification space I did the following cutting and pasting.

My question is : Is it correct way to do the identification? I have heard cutting is not a continuous operation. That's why I am confused. Could anyone give me some suggestions in this regard?

Thanks for your time.

Answer 1

This is absolutely correct, since cutting and pasting the shape does not actually change the space it illustrates. They're just different ways to visualize the same quotient space in our mind, called polygonal presentations of the space in some literature. The spaces having such a presentation are precisely the two-dimensional compact connected manifolds, or the surfaces. A chapter in Munkres' Topology gives the classification of the surfaces up to homeomorphism using this method.

Answer 2

This is a very clean visual argument, which any topologist 100 years ago would have accepted just as much as any topologist nowadays will. Of course, the crux to making such visual arguments is to understand how they, in principle, translate to a rigorous argument. If you've understood that principle, you can feel free to make such visual arguments for the rest of your life, but if you're a beginner in topology, you have to work out those principles at least once or twice.

Regarding the question about continuity, it is true that cutting a space usually changes its shape. However, when you're cutting, you are still drawing your spaces as quotients with certain identifications being made and those identifications (like identifying the two edges with four arrows on them in the second step) precisely undo the gluing, so you're simply writing down different ways of representing the same topological space as a quotient of different spaces.

Let me illustrate once how to make this precise: Say $X$ is the hexagon and $\sim$ is the equivalence relation that identifies the parallel edges on the boundary with one another. The topological space you're interested in is the quotient space $X/\sim$. Now, let $Y$ be the disjoint union of a triangle and a pentagon as drawn in the picture of your second step (but without the identifications). There is a clear map $Y\rightarrow X$, which is in fact a quotient map whose equivalence relation is given by identifying one edge of the triangle with one edge of the pentagon (the ones marked with four arrows in your picture). The composition of two quotient maps is a quotient map, so $Y\rightarrow X\rightarrow X/\sim$ is a quotient map. The equivalence relation on $Y$ describing the composite is precisely the one identifying all pairs of parallel edges on the boundary as in your second picture.

Thus, your first two illustrations show two different topological spaces ($X$ and $Y$), but with equivalence relations on them such that the quotients become one and the same. Now, to go from the second to the fourth picture, you repeat this process, but backwards. I recommend thinking that through. This is the symmetry between "cutting" and "gluing" and the heart of these arguments. The fact that the quotient space you are visually representing remains unchanged throughout is precisely why this argument works.