Exercises of unit2
1. Say true or false :
(a)  1 = {X,  , {a},{a, f}, {b, f}, {a, b, f}} is a
topology
on
X
=
{a,b,c,d,e,f}.
(
)
(b)  2 = {X,  ,{a, b, f}, {a, b, d } ,{a, b, d ,f } } is a
topology on X = {a, b, d, e, f}.
(
)
(c)  3 = {X, ,{b}, {a, b, c}, {d, e, f},{b, d, e, f}} is a
topology on X = {a, b, c, d, e, f}.
(
)
(d) if  4 = { X ,  , {a} , {b} , {b , d} ,{a, b, d} ,{a, b} }
is a topology on X = {a , b ,c , d} then { b,c,d} is
closed set.
(
)
(e) if  4 = { X ,  , {a} , {b} , {b , d} ,{a, b, d} ,{a, b} }
is a topology on X = {a , b ,c , d} then {a, b, c }
is open set.
(
)
(f) All topologies on X= {a} are discrete.
(
)
(g) No topologies we can construct on the set X= {
 }.
(
)
(h) We cannot define a proper topology on the set
of natural numbers N= {1, 2, 3,…}.
(
)
(i) Only three topologies we can construct on the
set X= {a, b}.
(
)
(j) All topologies on a nonempty set X must
contain a proper subsets of X.
(
)
6
2. If X = {a, b, c, d, e, f}.and  is the discrete
topology on X, which of the following statements
are true?
(a) X

(c) { }
(e) 
(d) 
X (
)
(f) { }



(o) {X}
 (
)
(



 (
)
(
)
X (
)




(
)
)
(j) {a}

X (
)
X (
)
(I) a
X
(
)
(
)
(n) {a}
 (
)
. (
)

(k) { }
(m) X
(b) {X}
)


)
 (


(g) {a}
(i) 
 (
 (
(h) a
)
(p) a



3. Let (X,  ) be any topological space. then the
intersection of any infinite number of members of
 is member of  .
(
)
4. Let R be the set all real numbers. Say true or
false.
T1 consists of R,  , and every interval (-n, n),
for any positive integer n is not a topology
on R.
(
)
(b)  2 consists of R,  , and every interval
(a)
[-n, n], for any positive integer n is a discrete
topology on R.
(
)
(c)  3 consists of R,  , and every interval

[-n,  ) for any positive integer n is a topology on R.
7
(
)
5. Let N be the set of all positive integers. Say true
or false
(a)  1 consists of N,  , and every set {1, 2, …., n},
for any positive integer n is a topology on N.
(
)
(b)  2 consists of N,  , and every set
{n, n + 1, …. }, for any positive integer n,
topology on N.
(
)
is not a
6. All possible topologies on X = {a, b} are 3.
(
)
7. Only three topologies on Y= {a, b, c} are quasidiscrete.
(
)
8. On Y= {a, b, c} there exist two topologies which
their union is also a topology on Y.
(
)
9. Let X be an infinite set and  a topology on X. If
every infinite subset of X is in  , then  is the
discrete topology.
(
)
10. Let R be the set of all real numbers. Then
(a) 1 consists of R,  , and every interval (a, b),
for a and b any real numbers with a < b; is not
a topology on R.
(
)
(b) 2 consists of R,  , and every interval (-r, r),
for r any positive real numbers; is a topology
on R.
(
)
8
(c) 3 consists of R,  , and every interval (-r, r),
for r any positive rational numbers; is a quasidiscrete topology on R.
(
)
(d) 4 consists of R,  , and every interval [-r, r],
for r any positive rational numbers is a
topology in which every open set is a closed
set.
(
)
(e) 5 consists of R,  , and every interval (-r, r),
for r any positive irrational numbers; is the
discrete space.
(
)
(f) 6 consists of R,  , and every interval [-r, r],
for r any positive irrational numbers; can not
to be a topology on R.
(
)
(g) 7 consists of R,  , and every interval [-r, r),
for r any positive real numbers is a topology

have {[r, r):r R } as abase.
(
)
(h) 8 consists of R,  , and every interval (-r, r],
for r any positive real numbers; is a topology
on R have no base.
(
)
(i) 9 consists of R,  , and every interval [-r, r],
and every interval (-r,r), for r any positive real
numbers is a union of two topologies on R.
(
)
(j) 10 consists of R,  , every interval [-n,n], and every
interval (-r, r), for n any positive integer and r any
positive real numbers is not a topology at all.
(
)
9
11. For any topology on the set X = { a, b, c, d }
there exist 32 subset of X that are clopen.
(
)
12. Let (X, T) be a topological space with the
property that every subset is closed. Then this
topology is the discrete space.
(
)
13. If (X, T) is a discrete space or an indiscrete
space then every open set is a clopen set.
(
)
14. Let X be an infinite set. There are no topologies
other than the indiscrete topology on X is finite.
(
)
15. Let X be an infinite set and T a topology on X
with the property that the only infinite subset of X
which is open is X itself. Is (X, T) necessarily an
indiscrete space?
(
)
16.
Let T be a topology on a set X such that T
consist of precisely four sets; that is, T = {X,  ,
A, B}, where A and B are nonempty distinct
proper subsets of X, then A and B must satisfy
exactly one of the following conditions:
(i) B = X / A; (ii) A  B;
(iii) B 
A.
(
)
17. Let  1 and  2 be two topologies on a set X. If  3
is defined by  3 = 1   2 then  3 is not necessarily
a topology on X.
(
)


10