Category:Disconnected Sets
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This category contains results about Disconnected Sets.
Definitions specific to this category can be found in Definitions/Disconnected Sets.
Let $T = \struct {S, \tau}$ 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 all of the following hold:
- $H \subseteq U \cup V$
- $H \cap U \cap V = \O$
- $U \cap H \ne \O$
- $V \cap H \ne \O$
Definition 3
$H$ is a disconnected set of $T$ if and only if there exist non-empty subsets $U$ and $V$ of $H$ such that all of the following hold:
- $H = U \cup V$
- no limit point of $U$ is an element of $V$
- no limit point of $V$ is an element of $U$.
Pages in category "Disconnected Sets"
This category contains only the following page.