...this must be seen as an artifact similar to the choice of coordinates in a vector space or manifold.* Finite polyhedra form a large universe of objects....homeomorphic, X ~ Y, if there are continuous maps f:X^Y and g:Y->X such that the composites fg = \Y and ' Compare H. Weyl, Philosopy of Mathematics and...

...Homeomorphisms DEFINITION 1.1. We say that topological spaces X and Y are homeomorphic if there exist continuous maps / : X — » Y and g : Y — » X such that the composites gof and fog are the identity maps of X and Y respectively; in short, go / = id and / og = id, where...

...and F t \A = f\A = g\B for all t £ [0,1]. Two topological spaces X, Y are called homotopy equivalent if there are continuous maps / : X —> Y and g : Y —> X such that the compositions fog and g o / are homotopic to the identity maps idy and id*, respectively. Then / is...