"A manifold with boundary has dimension at least 1" if it has a dimension and if it has nonempty boundary?

Question

My book is An Introduction to Manifolds by Loring W. Tu.

As can be found in the following bullet points

• 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

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

2. 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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Embedding photos:

Answer 1

I think Tu’s statement is fine:

A manifold, by definition, always has a dimension. Where are the charts going?

Usually when we say that a manifold “has boundary,” we mean that it has nonempty boundary.

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After looking at some of Tu’s (non-standard!) definitions, I think you’re correct. An accurate statement might be

If an $n$-dimensional manifold has nonempty boundary, then $n\ge 1$.

Answer 2

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.

Answer 3

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.