"A manifold with boundary has dimension at least 1" if it has a dimension and if it has nonempty boundary? My book is An Introduction to Manifolds by Loring W. Tu.
As can be found in the following bullet points


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*Can a topological manifold be non-connected and each component with different dimension?

*Is $[0,1) \cup \{2\}$ a manifold with boundary? My issue is the $2$.

*Understanding topological and manifold boundaries on the real line
we have that


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*Tu's manifolds with or without boundaries do not necessarily have (uniform) dimensions.

*Tu has considered manifolds to be manifolds with boundaries (with empty boundaries).
Question: For Definition 22.6 (see here and here), Tu says that "A manifold with boundary has dimension at least 1". Should this instead be "A manifold with boundary has dimension at least 1 if it has a dimension and if it has nonempty boundary" or "An $n-$manifold with boundary with non-empty boundary has $n \ge 1$" (Notice that the prefix "$n-$" precisely gives the manifold with boundary a dimension)?

Embedding photos:



Assuming sensible definitions, an alternative solution is to change the statement to the following:

A connected manifold with non-empty boundary has dimension at least 1

Edit: I rejected the suggested edit to change "manifold with non-empty boundary" to "manifold with boundary with non-empty boundary" because it does not add new information.  A manifold with non-empty boundary must be a manifold with boundary, or your definitions are nonsense.
