Let A be a set; let  be an indexed family of spaces; and let  be an indexed family of func...

Let A be a set; let  be an indexed family of spaces; and let  be an indexed family of functions fα : A → Xα.

(a) Show there is a unique coarsest topology  on A relative to which each of the functions fα is continuous.

(b) Let

and let  Show that  is a subbasis for

(c) Show that a map g : Y → A is continuous relative to  if and only if each

map fα o g is continuous.

(d) Let  be defined by the equation

let Z denote the subspace f(A) of the product space  Show that the image under f of each element of  is an open set of Z.