Unit 4
1. Let A be a subset of a topological space (X, T).
Then A dense in X if and only if every
neighbourhood of each point is X \ A intersects A
non-trivially.
(
)
2. Let A and B be subsets of a topological space (X,
T). Then
A B A B
(
)
3. Let (X, T) be a topological space. Then T is the
finite-closed topology on X if and only (a) (X, T) is
a T1-space, and (b) every infinite subset of X is
dense in X.
(
)
4. 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 (
)
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5. 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) =  .
(
)
6. If A is any subset of a topological space (X, T), the
int(A) = X  ( X  A) . (
)
7. Using Exercise 6 above, A is dense in (X, T) if and
only if int(X \ A) =  .(
)
8. Using the definition of int determine which the
following statements are true for arbitrary subsets
A1 and A2 of a topological space (X, T)?
(a) Int (A1 ∩ A2) = int (A1) ∩ int (A2), (
)
(b) Int (A1  A2) = int (A1)  int )A2), (
)
(c)
)
A A A A (
1
2
1
2
9. Let S be a dense subset of a topological space (X,
T). Then for every open subset U of X, S  U  U
(
)
10. Let S and T be dense subset of a space (X, T). if T
is also open then S ∩ T is dense in X. (
)
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