Disconnected Set

A Disconnected Set is a Set that is the union of two disjoint non-empty open sets.

• Example(s):
  • Uniformly Disconnected Julia Set (Fletcher & Vellis, 2020).
  • …
• Counter-Example(s):
  • Connected Set,
  • Closed Set,
  • Open Set.
• See: Disconnected Graph, Totally Disconnected Space, Disconnected Maximum Common Subgraph (dMCS), Felix Hausdorff, Topological Space, Subset, Empty Set, Closed Set, Clopen Set, Boundary (Topology), Separated Sets, Continuous_function, Frigyes Riesz, Maximal Element, Partition of a Set.

---

---

References

2020a

• (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Connected_space#Formal_definition Retrieved:2020-10-18.
  • A topological space X is said to be disconnected if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice.

    For a topological space X the following conditions are equivalent:

    1. X is connected, that is, it cannot be divided into two disjoint non-empty open sets.
    2. X cannot be divided into two disjoint non-empty closed sets.
    3. The only subsets of X which are both open and closed (clopen sets) are X and the empty set.
    4. The only subsets of X with empty boundary are X and the empty set.
    5. X cannot be written as the union of two non-empty separated sets (sets for which each is disjoint from the other's closure).
    6. All continuous functions from X to {0,1} are constant, where {0,1} is the two-point space endowed with the discrete topology.
  • Historically this modern formulation of the notion of connectedness (in terms of no partition of X into two separated sets) first appeared (independently) with N.J. Lennes, Frigyes Riesz, and Felix Hausdorff at the beginning of the 20th century. See for details.

2000b

• (ProofWiki, 2020) ⇒ https://proofwiki.org/wiki/Definition:Disconnected_(Topology)/Set Retrieved:2020-10-18.
  • QUOTE: Let $T=(S,\tau)$ be a topological space. Let $H\subseteq S$ be a non-empty subset of $S$.

    Definition 1

    $H$ is a disconnected set of $T$ if and only if it is not a connected set of $T$.

    Definition 2

    $H$ is a disconnected set of $T$ if and only if there exist open sets $U$ and $V$ of $T$ such that:

    $H\subseteq U\cup V$

    $H\cap U\cap V=\empty$

    $U \cap H \neq \empty$

    and

    $V \cap H \neq \empty$.

2020c

• (MathCS, 2020) ⇒ https://mathcs.org/analysis/reals/topo/connect.htmlt Retrieved:2020-10-18.
  • QUOTE: An open set $S$ is called disconnected if there are two open, non-empty sets $U$ and $V$ such that:

1. $U \cap V = 0$

2. $U \cup V = S$

A set $S$ (not necessarily open) is called disconnected if there are two open sets $U$ and $V$ such that

1. $(U \cap S)$ # $0$ and $(V \cap S)$ # $0$

2. $(U \cap S) \cap (V \cap S) = 0$

3. $(U \cap S) \cup (V \cap S) = S$

If S is not disconnected it is called connected.

2020d

• (Fletcher & Vellis, 2020) ⇒ Alastair N. Fletcher, and Vyron Vellis (2020). "On uniformly disconnected Julia sets". arXiv preprint arXiv:2004.01587.
  • QUOTE: It is well-known that the Julia set of a hyperbolic rational map is quasisymmetrically equivalent to the standard Cantor set. Using the uniformization theorem of David and Semmes, this result comes down to the fact that such a Julia set is both uniformly perfect and uniformly disconnected.

---