Embedding (simple definition)
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A smooth embedding of a smooth compact manifold into a smooth manifold is a smooth injective map such that is a monomorphism at each point. (See an equivalent alternative definition which works for non-compact manifolds and involves immersions. A smooth immersion is a smooth map such that is a monomorphism at each point. See an equivalent alternative definition.)
A map of a polyhedron is piecewise-linear (PL) if it is linear on each simplex of some smooth triangulation of . A PL embedding of a compact polyhedron into is a PL injective map .
A topological embedding of a compact subset into is a continuous injective map .