Is it true that continuity between topological spaces is what's responsible for "no tearing" and bijectivity is responsible for "no gluing"? The answer in this post: Is it correct to think about homeomorphisms as deformations? says that this is the truth.
However, a comment to that answer said that ""gluing together" implies the continuity of $f^{−1}$ (instead of bijectivity)"
I read in a comment from a separate post:What is the intuition behind homeomorphism, especially behind the geometrical notion of "gluing together"? which also says that ""gluing together" implies the continuity of $f^{−1} $"
I don't quite know which to be true. I can see how, for example, if two points are mapped to a single point, that would in a sense behave as gluing and would imply non-bijectivity.
However, I can also see gluing in a sense as reverse tearing, which would imply inverse continuity is responsible for gluing. Are they both somehow responsible?
Is it true that no tearing and no gluing is the reason why holes are preserved in homeomorphisms? If only bicontinuity is responsible for hole preservation, what role does bijectivity play in homeomorphisms?
Lastly, in a topological sense, can the reason for no tearing done by continuous functions be thought of as continuity preserving topological indistinguishability? Or, is preservation of connected spaces the only way of implying no tearing? I always read that continuity is about close points mapping to close points. I'll write out the indistinguishability preservation proof:
suppose $ f:(X, \mathcal{T}_x) \to (Y, \mathcal{T}_y)$ is continuous and that $f(x_1) \in U \in \mathcal{T}_y$
$\Rightarrow x_1 \in f^{-1}(U) \in \mathcal{T}_x$ by definition of pre-image and continuity
let $x_1$ and $x_2$ be indistinguishable
$\Rightarrow x_2 \in f^{-1}(U)$
$\Rightarrow f(x_2) \in U$
$\Rightarrow f(x_1)$ and $f(x_2)$ are indistinguishable
I believe this proof in a metric sense implies that points that are infinitesimally close/distance is zero to each other get mapped to points that are infinitesimally close/distance is zero to each other because if two points share every neighborhood, this must hold true for arbitrarily small neighborhoods. In topological spaces with no distance, it's best to refer to these points as indistinguishable rather than categorize by distance. This seems as a valid concept of no tearing because it sort of tells me that connectedness between points is preserved like a web or grid holding the points together, rather than connectedness between subsets. If there is a hole in a topological space, I don't necessarily see the space as having to equal disjoint subsets, which makes me wonder if this indistinguishability proof better describes preservation of tearing/gluing/holes verses the traditional preservation of connectedness of spaces by continuous functions.
Best Answer
I think the best way to think about this is if we take a map like $f : [0, 1) \to \mathbb{T}$ (the complex unit circle) given by $f(t) = e^{it}$, what this does is take the half open interval, and glue the two ends together. This is still bijective, and the forwards map is continuous, but the inverse map is not. The reason is that the forwards map took points that were far apart and made them arbitrarily close (gluing), so the inverse then takes those arbitrarily close points and makes them far apart. In this sense, we are not allowed to tear because of forwards continuity, and we are not allowed to glue (even bijectively!) because of inverse continuity. Tearing takes arbitrarily close points and makes them far, violating continuity, and gluing takes points which are far apart and make them arbitrarily close, which in the inverse, takes arbitrarily close points and makes them far, violating continuity.
The reason for the requirement that our map be a bijection is so that we can talk about the inverse at all! if our function is not bijective, $f^{-1}$ doesn't make sense.
Also note that this is a loose heuristic that is (in my opinion) only really helpful sufficiently nice subspaces of $\mathbb{R}^n$, so don't get too involved in doing it with gross arbitrary subspaces - think of grabbing some playdough and stretching/squishing it without tearing or gluing, nothing more. If anyone has a way to think about it with "gross" topological spaces, I'd appreciate a comment.