Let $X$ be a topological space, where

  • $X$ is Hausdorff.
  • $X = A \cup B$,
  • $A \cap B = \emptyset$,
  • $A$ is a topological $n$-manifold,
  • $B$ is a topological $(n-1)$-manifold,
  • $X = \overline{A}(X)$ (the closure of $A$ in $X$),
  • $B = \circ A(X) $ (the boundary of $A$ in $X$),

What are sufficient additional conditions for ensuring that $X$ is a manifold with boundary?

An example which shows that extra conditions are needed is $A = \{(x, \sin(1/x)) : x \in ]0, 1[\}$ and $B = \{(0, 0), (1, \sin(1)\}$ with the subspace topology of $\mathbb{R}^2$, i.e. $X$ is the topologist's sine. Then $X$ is not a manifold with boundary because $X$ is not locally connected at $(0, 0)$.

Although I'm interested in topological manifolds, if there is a simple answer for smooth manifolds, I'd be interested in that too, just to have more intuition.


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