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Connected Sets in Metric Spaces 📂MetricSpace

Connected Sets in Metric Spaces

Definition

If two subsets AA and BB of a metric space XX satisfy

AB=andAB= A \cap \overline{B}= \varnothing \quad \text{and} \quad \overline{A}\cap B= \varnothing

then AA and BB are said to be separated. In other words, there is no point of AA included in the closure of BB, and there is no point of BB included in the closure of AA. A subset EXE \subset X 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 ERE \subset \mathbb{R}, the following two propositions are equivalent.

(a) EE is a connected set.

(b) If x,yEx ,y\in E and x<z<yx<z<y, then zEz \in E.

Proof

  • (a) \Longrightarrow (b)

    The proof is by contrapositive. That is, if zEz\notin E, it will be shown that EE is a disconnected set. Assume zEz\notin E. Let’s say two subsets AzA_{z} and BzB_{z} as follows:

    Az=E(,z),Bz=E(z,)(1) A_{z}=E\cap (-\infty,z),\quad B_{z}=E\cap(z,\infty) \tag{1}

    Then

    E=AzBz E=A_{z}\cup B_{z}

    holds. Also, because of (1)(1), xAzx\in A_{z} and yBzy \in B_{z}, so both sets are non-empty. Similarly, because of (1)(1) AzBz=,AzBz= A_{z}\cap \overline{B_{z}}=\varnothing,\quad \overline{A_{z}}\cap B_{z}=\varnothing

    therefore, AzA_{z} and BzB_{z} are separated. Since EE can be represented as the union of two non-empty separated sets, by the definition, EE is a disconnected set.

  • (a) \Longleftarrow (b)

    Similarly, the proof is by contrapositive. That is, if EE is disconnected, then zEz\notin E. Assume EE is disconnected. Then, by the definition, there are two non-empty separated sets AA and BB that satisfy E=ABE=A \cup B. As AA and BB are non-empty, any xAx\in A and yBy\in B can be selected. Without loss of generality, let’s say x<yx<y. And, let’s say zz as follows:

    z=sup(A[x,y]) z =\sup (A\cap [x,y])

    Then, by the properties of closure1, the following holds:

    zA[x,y]A z \in \overline{A\cap [x,y]} \subset \overline{A}

    Since by assumption AA and BB are separated, therefore zBz \notin B. Now, let’s consider two cases:

    • case 1. zAz \notin A

      Given zAz \notin A, zBz \notin B, and E=ABE=A \cup B, then zEz\notin E.

    • case 2. zAz\in A

      Since by assumption AA and BB are separated, therefore zBz \notin \overline{B}. Hence, following the proof process, placing xx as zz gives:

      z<z1<y,z1B z<z_{1}<y,\quad z_{1}\notin B

      There exists z1z_{1} that satisfies this, and it also satisfies x<z1<yx<z_{1}<y and z1Ez_{1}\notin E.


  1. See Theorem 2 (2a), Theorem 4 ↩︎