
“real: chapter_04” — 2011/5/22 — 23:17 — page 21 — #21
Topological Aspects of the Real Line 4-21
that are valid for subsets of R and that can be extended (without any
modifications in the proofs) to the context of a metric space.
Theorem 4.3.7 Let (X , d) be a metric space.
(i) A set U ⊂ X is open if and only if its complement namely U
c
is
closed.
(ii) Arbitrary union of open sets is open.
(iii) Arbitrary intersection of closed sets is closed.
(iv) Finite intersection of open sets is open.
(v) Finite union of closed sets is closed.
(vi) For any subset U of X , U
◦
is always open. U ⊂ X is open if and
only if U = U
◦
and U
◦
is the largest open set contained