
“real: chapter_04” — 2011/5/22 — 23:17 — page7—#7
Topological Aspects of the Real Line 4-7
Examples 4.2.9
1. Note that an arbitrary intersection of open sets need not be open.
For example, consider the countable collection of open intervals
(−
1
n
,
1
n
) whose intersection is {0}, which is not open. Similarly,
using DeMorgan laws, it is easy to see that the countable union
of closed sets (−
1
n
,
1
n
)
c
, which is R \{0}, is not a closed set.
2. (A ∪ B)
◦
need not be equal to A
◦
∪ B
◦
. Indeed, let A = (0, 2],
B = (2, 3) be two subsets of
R so that (A ∪ B)
◦
= (0, 3). But
A
◦
∪ B
◦
= (0, 2) ∪ (2, 3) = (0, 3) \{2} = (A ∪ B)
◦
.
3.
A ∩ B need not be equal to
¯
A ∩
¯
B.InR consider