Connected Sets in Metric Spaces
Definition
If two subsets and of a metric space satisfy
then and are said to be separated. In other words, there is no point of included in the closure of , and there is no point of included in the closure of . A subset that cannot be represented as the union of two non-empty separated sets is said to be connected.
Reflecting on the definition above, we see that the concept of being “connected” was created to express a set that is “clearly represented as the union of overlapping sets.”
Theorem
For , the following two propositions are equivalent.
(a) is a connected set.
(b) If and , then .
Proof
(a) (b)
The proof is by contrapositive. That is, if , it will be shown that is a disconnected set. Assume . Let’s say two subsets and as follows:
Then
holds. Also, because of , and , so both sets are non-empty. Similarly, because of
therefore, and are separated. Since can be represented as the union of two non-empty separated sets, by the definition, is a disconnected set.
(a) (b)
Similarly, the proof is by contrapositive. That is, if is disconnected, then . Assume is disconnected. Then, by the definition, there are two non-empty separated sets and that satisfy . As and are non-empty, any and can be selected. Without loss of generality, let’s say . And, let’s say as follows:
Then, by the properties of closure1, the following holds:
Since by assumption and are separated, therefore . Now, let’s consider two cases:
case 1.
Given , , and , then .
case 2.
Since by assumption and are separated, therefore . Hence, following the proof process, placing as gives:
There exists that satisfies this, and it also satisfies and .
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See Theorem 2 (2a), Theorem 4 ↩︎