
“real: chapter_04” — 2011/5/22 — 23:17 — page 36 — #36
4-36 Real Analysis
Theorem 4.3.43 Every non-empty open subset of R
k
is at most a
countable disjoint union of open connected sets.
Proof By Theorem 4.3.40 (iv), any non-empty open subset U of
R
k
(considered as a metric space) can always be written as a disjoint union
of maximal connected sets namely its components. First we shall show
that these components are open. Indeed, if C is a component and x ∈ C,
we can always take a neighbourhood of x (a ball with center x radius
δ>0), which is completely contained in U . In as much as this ball
is connected (indeed, if B(x, δ) is disconnected, then we can