In mathematics, **topological complexity** of a topological space *X* (also denoted by TC(*X*)) is a topological invariant closely connected to the motion planning problem^{[ further explanation needed ]}, introduced by Michael Farber in 2003.

Let *X* be a topological space and be the space of all continuous paths in *X*. Define the projection by . The topological complexity is the minimal number *k* such that

- there exists an open cover of ,
- for each , there exists a local section

- The topological complexity: TC(
*X*) = 1 if and only if*X*is contractible. - The topological complexity of the sphere is 2 for
*n*odd and 3 for*n*even. For example, in the case of the circle , we may define a path between two points to be the geodesic between the points, if it is unique. Any pair of antipodal points can be connected by a counter-clockwise path. - If is the configuration space of
*n*distinct points in the Euclidean*m*-space, then

- The topological complexity of the Klein bottle is 4.
^{ [1] }

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- ↑ Cohen, Daniel C.; Vandembroucq, Lucile (2016). "Topological Complexity of the Klein bottle". arXiv: 1612.03133 [math.AT].

- Farber, M. (2003). "Topological complexity of motion planning".
*Discrete & Computational Geometry*.**29**(2). pp. 211–221. - Armindo Costa:
*Topological Complexity of Configuration Spaces*, Ph.D. Thesis, Durham University (2010), online

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