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.

   Solution to question 3

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?

   Solution to question 4

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

   Solution to question 5

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.

   Solution to question 6

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 ?

   Solution to question 7