General exams
Say true or false :
1. All topologies on X= {a} are discrete.
(
)
2. No topologies we can construct on the set X= {
 }.
(
)
3. We cannot define a proper topology on the set of
natural numbers N= {1, 2, 3,…}.
(
)
4. Only three topologies we can construct on the set
X= {a, b}.
(
)
5. All topologies on a nonempty set X must contain
a proper subsets of X.
(
)
6. 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
7. All possible topologies on X = {a, b} are 3.
(
)
8. Only three topologies on Y= {a, b, c} are quasidiscrete.
(
)
9. On Y= {a, b, c} there exist two topologies which
their union is also a topology on Y.
(
)
10. Let X be an infinite set and  a topology on X. If
every infinite subset of X is in  , then  is the
discrete topology.
(
)
11. 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.
(
)
(c) 3 consists of R,  , and every interval (-r, r),
for r any positive rational numbers; is a quasidiscrete topology on R.
(
)
12. If (X, T) is a discrete space or an indiscrete
space then every open set is a clopen set.
(
)
13. Let X be an infinite set. There are no topologies
other than the indiscrete topology on X is finite.
(
)
14. 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?
(
)
15. The topology T = { X,  , { 0 } }on the set X =
{0,1} is a  0 -space but not a  1 -space.
(
)
16. Each topology which is a  0 -space must be a
 1 -space.
(
)
17.
Let X be any infinite set. The countable-closed
topology is a  1 -space.
(
)
18. Let  1 and  2 be two topologies on a set X and
they are  0 - spaces. If
 is defined by  =  ∩ 
then  is not necessarily a  0 -topology on X.
3
3
1
2
3
(
)
19. If (X,  1 ) and (X,  2 ) are  1 -spaces, then (X,  1 ∩
 2 )is also a  1 -space.
(
)
20. A topological space (X, T) is said to be separable
if it has a dense subset which is countable. Then
(a) R with the usual topology is separable (
)
(b) A countable set with the discrete topology is
separable (
)
(c) A countable set with the finite-closed topology is
separable (
)
(d) (X, T) where X is finite is separable (
)
(e) (X, T) where T is finite is separable (
)
(f) An uncountable set with the discrete topology is
separable (
)
(g) An uncountable set with the finite-closed topology
is separable (
)
(h) A space (X, T) satisfying the second axiom of
accountability is separable (
)
21. Let (X, T) be any topological space and A any
subset of X. the largest open set contained A is called
the interior of A and is denoted by int (A). Then
(a) In R, int ([0, 1]) = (0, 1). (
)
(b) In R, int((3, 4)) = (3, 4). (
)
(c) If A is open in (X, T) then int {a} = {a }. (
)
(d) In R, int ({3}) =  .
(
)
(e) If (X, T) is an indiscrete space then, for all proper
subsets A of X, int (A) =  . (
)
(f) For every countable subset A of R, int (A) =  .
(
)
22. If A is any subset of a topological space (X, T), the
int(A) = X  ( X  A) . (
)
. Is the property of being a i  space i   0,1, 2, 2,3 is a
topological property.
(
)
 is
23. A space
x .
for every
(i)
(ii)
 D  O
 is
called a
D  space
G
(iii)

) If
)
if for every
and a closed set
false (
if x/ is a closed set
Then
is true (
a
open set
D  space
F
x   ,there
exists an
such that x  G
F
is
)
1  D
is true (
is compact and
)
F
is closed, then
F 
is
compact.
(
24. Let
)
1 ,......, m be
1 ...... m
(
compact subsets of a space
.
then
is also compact .
)
25.Compactness, sequentially compactness countably
compactness, weakly countably compactness
and
locally
properties.
(
)
compactness
are
topological