Topological complexity

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, introduced by Michael Farber in 2003. 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, introduced by Michael Farber in 2003. Let X be a topological space and P X = { γ : [ 0 , 1 ] → X } {displaystyle PX={gamma :, o ,X}} be the space of all continuous paths in X. Define the projection π : P X → X × X {displaystyle pi :PX o ,X imes X} by π ( γ ) = ( γ ( 0 ) , γ ( 1 ) ) {displaystyle pi (gamma )=(gamma (0),gamma (1))} . The topological complexity is the minimal number k such that