Manifold with boundary and topological boundary

Question

Why aren't the topological boundary of a manifold with boundary and the boundary of the manifold the same sets?

We defined the boundary of a manifold as follows:

$\partial M:= \lbrace p \in M : \exists (U,\varphi) \text{chart with} \varphi (p)=(0,...,x_n)^T \rbrace$

In contrast to the topological boundary being the closure minus the interior of M.

Answer 1

Take a simple example that lives in both situations. If $X$ is any topological space, the topological boundary of $X$ is $\varnothing$. Now, consider the case $X=[0,1]$ endowed with the inherited Euclidean subspace topology. This is also the simplest example of a $1$-manifold with boundary. As I said earlier, the topological boundary is $\varnothing$. However, the manifold boundary is the $2$-point set $\{0,1\}$. If you instead regard $[0,1]$ as a subset of the space $\mathbb{R}$ then its topological boundary is $\{0,1\}$, which is where your confusion lies.

Answer 2

The topological boundary equals the manifold boundary if and only if the manifold boundary is empty.

Explanation: Let $(X,T,A)$ be a manifold, i.e. let $T$ be a topology on $X$ and let $A$ be an atlas for $(X,T)$. The topological boundary - the boundary of $X$ with respect to $T$ - equals the manifold boundary - the boundary of $X$ with respect to $(T,A)$ - if and only if the manifold boundary is empty, since the topological boundary is always empty in this situation.

Answer 3

You are mixing apples with oranges.

The concept of topological boundary is a relative concept: given a topological space $X$ and a subset $Y \subset X$, the boundary of $Y$ relative to $X$ is, by definition, $\overline Y \cap \overline{X-Y}$. Very often "$X$" is given, and it is left out of the terminology, we narrow our attention to a specific subset of $X$, and we then think about the topological boundary of that subset; and this leads to great confusion which could be cleared up by emphasising that we are thinking about topological boundary of that subset relative to $X$.

Given a general topological space $X$, I suppose you could decide to define its topological boundary to be the boundary of $X$ relative to $X$, but that would be awfully boring, because it is always empty.