From Real to Complex Analysis by R. H. Dyer & D. E. Edmunds

Author:R. H. Dyer & D. E. Edmunds
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

and

The homotopy may be viewed as continuously deforming into through a family of paths with prescribed endpoints.

Definition 2.5.8

Let and be paths in a metric space such that . The product path is defined by

Similarly, if are paths in such that, for , , then the product path is defined by

Evidently is a path joining to ; likewise, is a path joining to .

Theorem 2.5.9

Let and be paths in a metric space ; suppose that and is defined. Then .

Proof

Let join to and join to since is defined, . As there exists a continuous map such that

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