▁sub man if old ▁which ▁is ▁often ▁useful . ▁Let ▁M ▁be ▁an ▁n - dimensional ▁manifold , ▁and ▁let ▁k ▁be ▁an ▁integer ▁such ▁that ▁ 0 ▁ ≤ ▁k ▁ ≤ ▁n . ▁A ▁k - dimensional ▁embedded ▁sub man if old ▁of ▁M ▁is ▁a ▁subset ▁S ▁ ⊂ ▁M ▁such ▁that ▁for ▁every ▁point ▁p ▁ ∈ ▁S ▁there ▁exists ▁a ▁chart ▁( U ▁ ⊂ ▁M , ▁ φ ▁: ▁U ▁→ ▁R n ) ▁containing ▁p ▁such ▁that ▁ φ ( S ▁ ∩ ▁U ) ▁is ▁the ▁intersection ▁of ▁a ▁k - dimensional ▁plane ▁with ▁ φ ( U ). ▁The ▁pairs ▁( S ▁ ∩ ▁U , ▁ φ | S ▁ ∩ ▁U ) ▁form ▁an ▁at las ▁for ▁the ▁differential ▁structure ▁on ▁S . ▁ ▁Alexander ' s ▁theorem ▁and ▁the ▁Jordan – S cho enf lies ▁theorem ▁are ▁good ▁examples ▁of ▁smooth ▁embed d ings . ▁ ▁Other ▁variations ▁ ▁There ▁are ▁some ▁other ▁variations ▁of ▁sub man if olds ▁used ▁in ▁the ▁literature . ▁A ▁neat ▁sub man if old ▁is ▁a ▁manifold ▁whose ▁boundary ▁agre es ▁with ▁the ▁boundary ▁of ▁the ▁entire ▁manifold . ▁Sh ar pe ▁( 1 9 9 7 ) ▁defines ▁a ▁type ▁of ▁sub man if old ▁which ▁lies ▁somewhere ▁between ▁an ▁embedded ▁sub man if old ▁and ▁an ▁imm ers ed ▁sub man if old . ▁ ▁Many ▁authors ▁define ▁topological ▁sub man if olds ▁also . ▁These ▁are ▁the ▁same ▁as ▁Cr ▁sub man