BRIEF ANSWERS FOR TEST 1

1. See pages 39 and 47 of the book. (Note: Other definitions may be correct, too.)

2. • Any infinite space with the cofinite topology. (Other examples may be correct, too.)
   • The 2-element space that is neither trivial nor discrete (the Sierpinski space). (Other examples may be correct, too.)

3. Suppose that U and V are dense open sets. Their intersection (U \intersect V) = Wis open since the intersection of any two open sets are open. To prove that W is dense, we must show that the intersection of W with any nonempty open set O is nonempty. But (W\intersect O) = ((U\intersect V)\intersect O) = (U\intersect (V\intersect O)). The set (V\intersect O) is nonempty and open, since V is dense and all sets are open. Hence it has nonempty intersection with the dense set U. This proves that (W \intersect O) is nonempty.

4. To prove that f is continuous it suffices to prove that the inverse image of a closed set is closed. The inverse image of the whole space is the whole space, which is closed. Any other closed set is finite, say C = {c_1,...,c_k}. Then f^(-1)(C) is the union of f^(-1)(c_1), ..., and f^(-1)(c_k). Since each of these sets is finite, and there are finitely many of them, f^(-1)(C) is finite and hence closed.