Boundary of a connected manifold is connected?

Question

I assume the answer to this question is simple, but I can't find any references:

Let $X$ be a topological space, and $M$ be a (path-)connected component of a manifold in $X$ with the same dimension as $X$. Then, is the boundary of $M$ also (path-)connected?

Answer 1

The boundary need not be (path-)connected.

Consider $X = \mathbb{R}$ with its standard topology and $M = [0, 1]$. Note that $\partial M = \{0, 1\}$ which is not connected or path-connected.