BRIEF ANSWERS FOR TEST 2

1. See pages 68 and 77 of the book. (Note: Other definitions may be correct, too.)

2. • The topologist's sine curve, or the space on the cover of the textbook. (Other examples may be correct, too.)

   • {1/n | n = 1, 2, 3, ...} union {0}. (Other examples may be correct, too.)

3. No.

   If Y is compact, then so is Y x Y (by Tychonoff's Theorem). If Z x Z were homeomorphic to Y x Y, then it would also be compact. But if Z x Z were compact, then Z would also be compact since it would be a continuous image of a compact space. Since Z is not compact, the answer is No. (A similar argument works using connectivity in place of compactness.)

4. Choose any point p = (x,y) in (X x X) - D. Necessarily x is not equal to y. Since X is Hausdorff there exist disjoint open sets U and V with x in U, y in V. The set U x V is an open set of X x X containing (x,y) which does not intersect the diagonal (since U is disjoint from V). Hence (x,y) is an interior point of (X x X) - D.