the notion of symmetry is usually captured by considering a group action of formula_2 on formula_3 and formula_4 and demanding that formula_5 is equivariant under this action, so that formula_6 for all formula_7, a property usually denoted by formula_8 . heuristically speaking, standard topology views two spaces as equivalent "up to deformation," but equivariant topology considers spaces equivalent up to deformation so long as it pays attention to any symmetry possessed by both spaces. a famous theorem of equivariant topology is the borsuk–ulam theorem , which asserts that every formula_9-equivariant map formula_10 necessarily vanishes.