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This article defines a property over pairs of a topological space and a subspace, or equivalently, properties over subspace embeddings (viz, subsets) in topological spaces


A subspace of a manifold is termed a submanifold if it is a manifold with the subspace property.

A submanifold may in general have smaller dimension than the whole manifold. The difference between the dimension of the whole manifold and that of the submanifold is termed the codimension of the submanifold.


  • Every open subset of a manifold is a submanifold of the same dimension
  • If any non-open subset of a connected manifold is a submanifold, the dimension must be smaller