sentences:  Now, for each "i", take "Xi" to be "Ai" with the index "i" itself tacked on (renaming the indices using the disjoint union if necessary, we may assume that "i" is not a member of "Ai", so simply take "Xi" = "Ai" ∪ {"i"}).Now define the cartesian product
  
  along with the natural projection maps "πi" which take a member of "X" to its "i"th term.We give each "Xi" the topology whose open sets are the cofinite subsets of "Xi", plus the empty set (the cofinite topology) "and" the singleton {"i"}.