Metric and Topological Spaces

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## Exercises 5

1. Find the interior and closure of Q in R when R has:
1. the usual topology
2. the discrete topology
3. the trivial topology
4. the cofinite topology [finite sets are closed]
5. the co-countable topology [countable sets are closed]
6. the topology in which intervals (x, ) are open

Solution to question 1

2. Let N have the topology of Exercises 4, Question 8.
(This is the subspace topology as a subset of R with the topology of Question 1(vi) above.)
Find the interior and closure of the sets:
1. {36, 42, 48}
2. the set of even integers

Solution to question 2

3. A subset A of a topological space X is said to be dense in X if the closure of A is X.
(i) Prove that both Q and R - Q are dense in R with the usual topology.
(ii) Find all the dense subsets of N with the topology of the last question.

4. Let A, B be any subsets of a topological space. Show that cl(A B) cl(A) cl(B) where cl indicates the closure.
Give an example to show that equality might not hold.
Prove that int(A) int(B) = int(A B) and that int(A) int(B) int(A B) where int indicates the interior.
Can this last inclusion ever be proper?

5. Is the usual topology on R stronger or weaker than the cofinite topology ?

6. Consider R with the cofinite topology. Show that the subspace topology on any finite subset of R is the discrete topology. Show that the subspace topology on the subset Z is not discrete.

7. Show that there are four different topologies on the set {a, b}. How many of them are non-homeomorphic ?
Show that there are 29 different topologies on the set {a, b, c}. How many of them are non-homeomorphic ?

SOLUTIONS TO WHOLE SET
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JOC February 2004