Problem 8: Given (X, Tx) and (Y, Ty) 2 topological spaces. Recall that the cartesian product X Y = {(x, y)|x e X, y E Y} 1. Consider the collection B of subsets of X x Y defined by A E B if and only if there exist some open sets U Tx and V E Ty such that A = U V Show that B is a basis for a topology on X x Y. The topology generated by B is called the Product Topology on X x Y. We will denote it by TxxY and that by definition of a basis for a topol- ogy, an open set TE Txxy is a union of elements of B. 2. Consider the projection map A : (X Y,TxxY) (X,Tx) defined by T(x, y) = x.Show that r is a continuous map. (Note that the projection map onto (Y, Ty) is also a continuous map.) 3. Consider the map f : (X x Y, TxxY) (Y X, Tyxx) defined by f(x, y) = (y, x). Show that f is a continuous map. 4. Consider the maps fi : (X Y, TxxY) (X, Tx) and f2 : (X x Y, TxxY) (Y, TY). Show that the map g : (X x Y,TxxY) (X x Y, TxxY) defined by g(x, y) = (fi(x, y), f2(x, y)) is continuous if and only if both fi and f2 are continuous. (you may use results from the previous questions) Problem 10: Suppose f : (X, Tx) (Y, Ty) is continuous, surjective, that the image through f of any closed subset of X is a closed subset of Y and that for all y Y, f-'({y}) is compact subset of X. (so f satisfies 4 different properties). Show that if (Y, Ty) is compact then (X, Tx) is also compact. (Hint: if U an open set of X containing f-'({y}), then there is an open set W of Y containing y such that f"(W) is included in U.) Problem 8: Given (X, Tx) and (Y, Ty) 2 topological spaces. Recall that the cartesian product X Y = {(x, y)|x e X, y E Y} 1. Consider the collection B of subsets of X x Y defined by A E B if and only if there exist some open sets U Tx and V E Ty such that A = U V Show that B is a basis for a topology on X x Y. The topology generated by B is called the Product Topology on X x Y. We will denote it by TxxY and that by definition of a basis for a topol- ogy, an open set TE Txxy is a union of elements of B. 2. Consider the projection map A : (X Y,TxxY) (X,Tx) defined by T(x, y) = x.Show that r is a continuous map. (Note that the projection map onto (Y, Ty) is also a continuous map.) 3. Consider the map f : (X x Y, TxxY) (Y X, Tyxx) defined by f(x, y) = (y, x). Show that f is a continuous map. 4. Consider the maps fi : (X Y, TxxY) (X, Tx) and f2 : (X x Y, TxxY) (Y, TY). Show that the map g : (X x Y,TxxY) (X x Y, TxxY) defined by g(x, y) = (fi(x, y), f2(x, y)) is continuous if and only if both fi and f2 are continuous. (you may use results from the previous questions) Problem 10: Suppose f : (X, Tx) (Y, Ty) is continuous, surjective, that the image through f of any closed subset of X is a closed subset of Y and that for all y Y, f-'({y}) is compact subset of X. (so f satisfies 4 different properties). Show that if (Y, Ty) is compact then (X, Tx) is also compact. (Hint: if U an open set of X containing f-'({y}), then there is an open set W of Y containing y such that f"(W) is included in U.)