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Given a manifold <jats:italic>M<\/jats:italic>, a simplicial complex\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathscr {A}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>A<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and a map <jats:italic>H<\/jats:italic> from the underlying space of <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathscr {A}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>A<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> to\u00a0<jats:italic>M<\/jats:italic>, our criteria are presented in local coordinate charts for\u00a0<jats:italic>M<\/jats:italic>, and ensure that <jats:italic>H<\/jats:italic> is a homeomorphism.\n\n These criteria do not require a differentiable structure, or even an explicit metric on\u00a0<jats:italic>M<\/jats:italic>. No Delaunay property of <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathscr {A}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>A<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. 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