Talk:Boundary parallel

The text {{tqq|In mathematics, a closed n-manifold N embedded in an (n + 1)-manifold M is boundary parallel (or ∂-parallel, or peripheral) if there is an isotopy of N onto a boundary component of M.}} is unclear for several reasons:

• Boundary (topology) pertains to a subset of a topological space; the topological boundary of the entire space is the empty set. Should that be Manifold#Boundary and interior?
• Homotopy#Isotopy pertains to a pair of functions
• The pair of links boundary component violates WP:SOB

I considered {{tqq|In mathematics, a closed n-manifold N embedded in an (n + 1)-manifold with boundary M is boundary parallel (or ∂-parallel, or peripheral) if there is an isotopy of N onto a component of M{{`}}s boundary.}}, but that seems stilted and doesn't address the second issue.

How about {{tqq|In mathematics, an embedding $f\backslash colon\; N\; \backslash rightarrow\; M$ of a closed {{mvar|n}}-manifold {{mvar|N}} in an ({{mvar|n}} + 1)-manifold with boundary {{mvar|M}} is boundary parallel (or ∂-parallel, or peripheral) if there is an embedding $g\backslash colon\; N\; \backslash rightarrow\; C$ of {{mvar|N}} onto a component {{mvar|C}} of {{mvar|M}}{{`}}s boundary and {{mvar|f}} is isotopic to {{mvar|g}}.}}

Is the concept of components of the boundary of a manifold with boundary important enough to warrant a section or anchor somewhere? Note that boundary component links to the wrong definition and probably should be a DAB page. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 08:23, 12 June 2025 (UTC) -- Revised 13:12, 12 June 2025 (UTC)

:Unless someone objects I'll go with {{tqq|In mathematics, an embedding $f\backslash colon\; N\; \backslash rightarrow\; M$ of a closed {{mvar|n}}-manifold {{mvar|N}} in an ({{mvar|n}} + 1)-manifold with boundary {{mvar|M}} is boundary parallel (or ∂-parallel, or peripheral) if there is an embedding $g\backslash colon\; N\; \backslash rightarrow\; C$ of {{mvar|N}} onto a component {{mvar|C}} of {{mvar|M}}{{`}}s boundary and {{mvar|f}} is isotopic to {{mvar|g}}.}} -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:19, 25 June 2025 (UTC)