"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)?

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I would not say that a manifold could be dimensionless.  A manifold consists of connected components, each of which has a dimension. As for the statement in question, a more accurate phrasing would be
"If an n-manifold has nonempty boundary, then $n \ge 1$"
or
"A connected manifold with nonempty boundary has dimension at least 1"
as was pointed out above by various commentators.
