- the usual topology
- the discrete topology
- the trivial topology
- the cofinite topology [finite sets are closed]
- the co-countable topology [countable sets are closed]
- the topology in which intervals [x, ] are open
Solution to question 1
[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:
- {36, 42, 48}
- the set of even integers
Solution to question 2
[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
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
Solution to question 5
Solution to question 6
Show that there are 29 different topologies on the set {a, b, c}. How many of them are non-homeomorphic ?
Solution to question 7