110.413 INTRODUCTION TO TOPLOGY 8 December 1998

Final Examination

50 minutes. Open book. 60 points, 20 per question. Begin each question on a new page and number it clearly in the margin.

1. Consider the projection π_{1}: X × Y → X to the first factor.

(a) If Y is compact, show that π_{1} is a closed map, i.e. that π_{1}(C) is closed whenever C ⊂ X × Y is closed.

(b) What happens if Y is not compact?

2. Let A be a closed subspace of R^{2}.

(a) If A is connected, is it necessarily true that the interior of A is connected?

(b) If A is connected, is it necessarily true that the boundary of A is connected?

(c) What about the converses of (a) and (b)?

3. What can you say about the fundamental groups of the following spaces: (a) The space S^{1} × R with one point deleted;

(b) The space R^{3} with the line x = y = z deleted;

(c) The union of the three axes in R^{3};

(d) R × Z.

YOU MAY RETAIN THIS QUESTION SHEET JMB