
“real: chapter_04” — 2011/5/22 — 23:17 — page 31 — #31
Topological Aspects of the Real Line 4-31
we can get a finite sub-cover for S using the finite number of these
balls of radius
δ
3
(which covers S) and the corresponding G
α
s. Hence
S is compact.
In the following we shall obtain certain properties of compact
subsets in a metric space, which will be useful later.
Theorem 4.3.31 Let (X , d) be a metric space. If A ⊂ X is compact
and B ⊂ X is closed with A ∩ B =∅, then d(A, B)>0.
Proof Since d(A, B ) ≥ 0 always, we shall assume d(A, B) = 0 and
deduce a contradiction to the hypothesis. Indeed if d(A, B) = 0 then
from the definition and Theorem 4.2.18,