### Topology Exercises

You may work alone or in a group of up to 3 people and turn in 1 per group. Make copies of your work so that you are prepared to present any of the exercises after you turn it in.
1. Munkres p. 20 #2 part c.

2. Mendelson p. 75 #6

3. Assume that X is a set with topologies T1 and T2 on X. Is the union of the two topologies necessarily a topology on X? If so prove the statement, but if not, give a counterexample.

4. Prove that [a,b) with reals a < b is not open in the standard topology on the reals. Is (a,b) open in the lower limit topology on the reals? Prove your answer. Is the lower limit topology finer, coarser, or not comparable with the standard topology? Explain.

5. Munkres p. 83 #7. As in the instructions, determine containments. Do not write out complete proofs of your results, but if T1 C T2, then do informally justify why opens in T1 are open in T2, and for strict containment T1 C T2, do list an open set in T2 that is not open in T1.

Grad Problem Munkres p. 83 #4