Clarification on Gluing Manifolds and Hausdorff Condition

Question

I'm reading about the process of gluing manifolds and have come across the following example in my studies:

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My question

I understand that we are considering a collection of manifolds and gluing them together using homeomorphisms. However, I am a bit unclear on what $U_{i,j}$ is first of all... Why $U_{i,j} \subset M_i$? Like what's their indexing system?

Any assistance would be appreciated!

Answer 1

If perhaps one goes through what is "given" in this definition, one thing at a time, things will sort themselves out.

To start with:

We are given $M_i$ ($1 \le i \le n$), a finite collection of manifolds.

The intuition here is that for each $1 \le i \le n$ and $1 \le j \le n$, a subset of $M_i$ is to be glued to a subset of $M_j$. The next portion of the givens specifies these subsets:

For each $1 \le i \le n$ and $1 \le j \le n$ we are given an open subset $U_{i,j} \subset M_i$.

The intuition is that $U_{i,j} \subset M_i$ and $U_{j,i} \subset M_j$ are the subsets that are to be glued together, but how exactly this is to be done is not yet clear. The final portion of the givens specifies exactly how these gluings are to be carried out:

For each $1 \le i \le n$ and $1 \le j \le n$ we are given a homeomorphism $f_{i,j} : U_{i,j} \to U_{j,i}$.

The intention here is that each point $x \in U_{i,j}$ is to be glued to the point $f_{i,j}(x) \in U_{j,i}$. One now has enough information to form the quotient space under this gluing.

The rest of the definition lays out constraints that all these givens have to satisfy in order for everything to work out nicely, namely in order for the quotient space to be a manifold. That must still be proved, and there's still a fair amount of details to the proof that seem to be shoved under the rug. And, as said in the comments, one or two of the constraints seem to have been overlooked.

I would also add a formal, set-theoretic constraint that the manifolds $M_1,...,M_n$ be pairwise disjoint to start with and that their disjoint union $M_1 \sqcup \cdots \sqcup M_n$ be assigned the disjoint union topology. Or, to put it another way, once the manifolds $M_1,...,M_n$ are given we form their coproduct, and the rest of the definition is specifying a quotient of their coproduct.