Use of the Term "Topological Invariance" in Lee's Introduction to Smooth Manifolds

Question

I'm reading through Lee's Introduction to Smooth Manifolds, and Theorem 1.37, which asserts that the interior and boundary of a manifold with boundary are disjoint, is titled "Topological Invariance of the Boundary." I'm having trouble seeing why this name fits. Isn't a topological invariant a property that is not affected by homeomorphisms? What's the connection between that idea and this theorem?

Here's my attempt at a guess: A homeomorphism from one manifold with boundary to another must map interior points to interior points, and this theorem ensures that it will map boundary points to boundary points as well.

Answer 1

It seems that you are thinking of a "topological invariant" as a mathematical statement which is either true or false for any topological space, and which is preserved under homeomorphism, such as the statements "$X$ is compact" and "$X$ has seven connected components".

But there is a broader notion of a "topological invariant", which can be understood with a tiny bit of category theory: a topological invariant is a kind of function, called a "functor" in category theory, which inputs a topological space and outputs some other kind of object, such that if the inputs are homeomorphic then the outputs are isomorphisms of that other kind of object.

With that in mind, the theorem you are looking for is this:

Let $M$ and $N$ be manifolds with boundary, and denote their boundaries as $\partial M$ and $\partial N$ and the interiors as $\text{int}(M)$ and $\text{int}(M)$. For any homeomorphism $f : M \to N$ we have $f(\partial M) = \partial N$, and $f(\text{int}(M)) = \text{int}(N)$.

The boundary operator, and the interior operator, are therefore functors whose outputs are topological spaces (as an exercise, the restricted maps $f \mid \partial M : \partial M \to \partial M$ and $f \mid \text{int}(M) : \text{int}(M) \to \text{int}(M)$ are both homeomorphisms).

I don't know how much topology you know, but here are a few other examples of topological invariants in this sense: the fundamental group; the set of connected components; the set of path components.