Analysis MOC

Continuous path

Given a Topological space a continuous path or path in is a continuous function , where . Iff is also a Embedding it is called an arc. topology A continuous path with the same start and endpoints is a Continuous loop.

Algebra

The set of paths may be made into a Magmoid with the concatenation operation. Let and . Then their concatenation is defined as

Additionally, we have the involution of reverse path traversal: For its reverse path is given by

Clearly defines a functor from Category of topological spaces to Category of magmoids Of more importance are the Category of paths and Fundamental groupoid, which are quotients modulo traversal and homotopy of paths respectively.

Properties

• A path is just a homotopy of maps from the Single point space, i.e. of constant maps.

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