Equivalence of Definitions of Disconnected Space
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Theorem
The following definitions of the concept of Disconnected Space are equivalent:
Let $T = \struct {S, \tau}$ be a topological space.
Definition $1$
$T$ is disconnected if and only if $T$ is not connected.
Definition $2$
$T$ is disconnected if and only if there exist non-empty open sets $U, V \in \tau$ such that:
- $S = U \cup V$
- $U \cap V = \O$
That is, if there exists a partition of $S$ into open sets of $T$.
Proof
By definition $1$, $T$ is disconnected if and only if $T$ is not connected.
By definition of connected topological space:
$T$ is connected if and only if:
Hence precisely a disconnected space by definition $2$.
$\blacksquare$