ANALYSIS II
Metric Spaces: Connectedness
Defn. A disconnection of a set A in a metric space (X,d) consists of two nonempty sets A1, A2
whose disjoint union is A and each is open relative to A. A set is said to be connected if it does
not have any disconnections.
Example. The set (0,1/2) ∪(1/2,1) is disconnected in the real number system.
Theorem. Each interval (open, closed, half-open) I in the real number system is a connected set.
Pf. Let A1,A2 be a disconnection for I. Let aj ∈ Aj, j = 1,2. We may assume WLOG that a1 < a2,
otherwise relabel A1 and A2. Consider E1: = {x ∈ A1 | x ≤ a2 }, then E1 is nonempty and bounded from
above. Let a: = supE1. But a1 ≤ a ≤ a2 implies a ∈ I since I is an interval. First note that by the lemma to
the least upper bound property either a ∈ A1 or a is a limit point of A1. In either case, a ∈ A1 since A1 is
closed relative to I. Since A1 is also open relative to the interval I, then there is an ε > 0 so that Nε(a) ∈
A1. But then a+ε/2 ∈ A1 and is less than a2, which contradicts that a is the sup of E1. [¯]
Theorem. If A is a connected subset of real numbers (with the standard metric), then A is an
interval.
Pf. Otherwise, there would be a1 < a < a2, with aj ∈ A and a does not belong to A. But then O1 : =
(-∞,a)∩A and O2 : = (a,∞)∩A form a disconnection of A. [¯]
Note. Each open subset of IR is the countable disjoint union of open intervals. This is seen by
looking at open components (maximal connected sets) and recalling that each open interval
contains a rational. Relatively (with respect to A ⊆ IR) open sets are just restrictions of these.
Theorem. The continuous image of a connected set is connected.
Pf. If C is a connected set in a metric space X and there is a disconnection of the image f(C), then it can
be `drawn back' to form a disconnection of C : if { O1, O2 } forms a disconnection for f(C), then {
f-1(O1),f-1(O2) } forms a disconnection for C. [¯]
Corollary. (Intermediate Value Theorem) Suppose f is a real-valued function which is
continuous on an interval I. If a1, a2 ∈ I and y is a number between f(a1) and f(a2), then there
exists a between a1 and a2 such that f(a) = y.
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Connectedness in Metric Space http://www.math.sc.edu/~sharpley/math555/Lectures/MetricSpaceConnectedness.html
