Free actions of finite groups on Hausdorff spaces The notion of a group action on a topological space is defined on page 54 of Munkres For our purposes it will suffice to take a group and to view it as topological groups with respect to the discrete topology If G is such a group and X is a topological space the group action itself is given by a continuous mapping G X X with g x usually abbreviated to g x or gx such that 1 x x for all x and gh x g h x for all g h and x One can then define an equivalence relation on X by stipulating that y x if and only if y g x for some g G and the quotient space with respect to this relation is called the orbit space of the group action and written X G The equivalence classes often called the orbits of the group action are the sets G x of all points of the form g x where g G Note that if X is Hausdorff and G is finite then G x is a closed compact in fact finite subset for each x X PROPOSITION Suppose that G is a finite group acting on a Hausdorff space X Then the orbit space X G is Hausdorff Proof One fast way of proving this result is to use the following fact about Hausdorff spaces If E and F are disjoint compact subsets of a Hausdorff space X then there are disjoint open subsets U and V such that E U and F V The proof of this is a fairly straightforward exercise Note that this result applies if E and F are both finite Let x y X be such that G x 6 G y and apply the result in the preceding paragraph to x and G y to obtain disjoint open neighborhoods U 0 and V0 of these compact subsets Let V g g V0 where g runs through all the elements of G The V is a G invariant open neighborhood of G y and U0 V Let U g g U0 Then U and V are disjoint G invariant open neighborhoods of G x and G y respectively Let X X G denote the quotient projection We claim that U and V are disjoint open neighborhoods of x and y respectively The two sets in question are open in the quotient topology because 1 U U 1 V V and U and V are open in X Likewise U and V are disjoint for if z lies in their intersection then G z U V and we know that the latter is empty If we are given a group action as above and A is a subset of X then for a given g G it is customary to let g A the translate of A by g be the set g A this is the set of all points expressible as g a for the fixed g and some a A Definition We shall say that a group action as above is a free action or G acts freely if for every x X the only solution to the equation g x x is the trivial solutions for which g 1 If X S 2 as above and G is the order two subgroup 1 of the real numbers with respect to multiplication then scalar multiplication defines a free action of G on S 2 and the quotient space is just RP2 Of course there are also similar examples for which 2 is replaced by an arbitrary positive integer n and in this case the quotient space S n 1 is called real projective n space Some links to the motivation for this definition are given in the following course directory document projective spaces links pdf Further information about the relationship is contained in Exercises V 1 2 and V 1 3 on page 12 of the following document 1 http math ucr edu res math205A gentopexercises2008 pdf Solutions are given on page 6 of the document math205Asolutions4 pdf in the same directory The next result implies that the orbit space projections S n RPn are covering space projections THEOREM Let G be a finite group which acts freely on the Hausdorff topological space X and let X X G denote the orbit space projection Then is a covering space projection Proof Let x X be arbitrary and let g 6 1 in G Then there are open neighborhoods U 0 g of x and V0 g of g x that are disjoint If we let W g U g g 1 V g is another open set containing x while g W g is an open set containing g x and we have W g g W g Let W W h h6 1 so that W is an open set containing x We claim that if g1 6 g2 then g1 W g2 W If we know this then it will follow immediately that W is an open set in X G whose inverse image is the open subset of X given by g g W This and the definition of the quotient topology imply that W is an evenly covered open neighborhood of x and therefore it will follow that is a covering space projection Thus it remains to prove the statement in the first sentence of the preceding paragraph Note first that it will suffice to prove this in the special case where g 1 1 assuming we know this in the general case we then have g1 W g 2 W g1 W g1 1 g2 W and the coefficient of g1 on the right hand side is empty by the special case when g 1 1 and the fact that g1 6 g2 implies 1 6 g1 1 g2 But if g 6 1 then we have W g W W g g W g and we know that the latter is empty by construction Therefore W g W and as noted before this completes the proof 2