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In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions.
Similar to homotopy classes, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them. Regular homotopy for immersions is similar to isotopy of embeddings: they are both restricted types of homotopies. Stated another way, two continuous functions f , g : M → N {\displaystyle f,g:M\to N} are homotopic if they represent points in the same path-components of the mapping space C {\displaystyle C} , given the compact-open topology. The space of immersions is the subspace of C {\displaystyle C} consisting of immersions, denoted by Imm {\displaystyle \operatorname {Imm} }. Two immersions f , g : M → N {\displaystyle f,g:M\to N} are regularly homotopic if they represent points in the same path-component of Imm {\displaystyle \operatorname {Imm} }.