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. #m/def/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 to
Of more importance are the Category of paths and Fundamental groupoid,
which are quotients modulo traversal and homotopy of paths respectively.