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- Find the interior and closure of
**Q**in**R**when**R**has:

- 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 - the usual topology
- 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:

- {36, 42, 48}

- the set of even integers

Solution to question 2 - {36, 42, 48}
- 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. - 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? - Is the usual topology on
**R**stronger or weaker than the cofinite topology ? - 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. - 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 ?

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