Page 33 - DMTH503_TOPOLOGY

Unit 1: Topological Spaces

3. Let (X, T) be any topological space. Verify that the intersection of any finite number of Notes
member of T is a member of T.
4. List all possible topologies on the following sets:

(a) X = {a, b}; (b) Y = {a, b, c}
5. Let X be an infinite set and T a topology on X. If every infinite subset of X is in T, prove that
T is the discrete topology.

6. Let (X, T) be a topological space with the property that every subset is closed. Prove that it
is a discrete space.
7. Consider the topological space (X, T) where the set X = {a, b, c, d, e}, the topology T = {X, ,
{a}, {c, d}, {a, c, d}, {b, c, d, e}}, and A = {a, b, c}. Then b, d, and e are limit points of A but a and
c are not limit points of A.
8. Let X = {a, b, c, d, e} and T = {X, , {a}, {c, d}, {a, c, d}, {b, c, d, e}} show that { b } = {b, c}, { a, c }
= X, and { b, d } = {b, c, d, e}.
9. Let X = {a, b, c, d, e, f} and
T = {X, , {a}, {c, d}, {a, c, d}, {b, c, d, e, f}},
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(a) Find all the limit points of the following set:

(i) {a},
(ii) {b, c},
(iii) {a, c, d},
(iv) {b, d, e, f},
(b) Hence, find the closure of each of the above sets.

10. (a) Let A and B be subsets of a topological space (X, T). Prove carefully that A  B 
A  B .
(b) Give an example in which A  B  A  B .

11. Let S be a dense subset of a topological space (X, T). Prove that for every open subset  of
X, S   = .
12. Let E be a non-empty subset of a topological space (X, T). Show that E = E  d (E), where
d (E) is derived set of E.

13. Define interior operator. Explain how can this operator be used to define a topology on a
set X.
14. Prove that A subset of topological space is open iff it is nhd of each of its points.
15. (a) Show that A° is the largest open set contained in A.

(b) Show that the set of all cluster points of set in a topological space is closed.
16. The union of two topologies for a set X is not necessarily a topology for X. Prove it.
17. Let X be a topological space. Let A  X. Then prove that A  A is closed set.
18. Show that A  D(A) is a closed set. Also show that A  D(A) is the smallest closed subset
of X containing A.
19. In a topological space, prove that (X – A)° = X – A. Int A = ( A). Hence deduce, that
A° = ( A).

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