Non-Trivial Discrete Space is not Connected
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Theorem
Let $T = \struct {S, \tau}$ be a non-trivial discrete topological space.
Then $T$ is not connected.
Thus also:
Corollary 1
$T$ is not path-connected.
Corollary 2
$T$ is not arc-connected.
Corollary 3
$T$ is not irreducible.
Corollary 4
$T$ is not ultraconnected.
Proof
Let $T = \struct {S, \tau}$ be a non-trivial discrete space.
Let $a \in S$.
Let $A = \set a$ and $B = \relcomp S {\set a}$, where $\relcomp S {\set a}$ is the complement of $A$ in $S$.
As $T$ is not trivial, $B \ne \O$.
Then from Set in Discrete Topology is Clopen, $A$ and $B$ are both open.
So $A \mid B$ is a separation of $S$.
It follows, by definition, that $T$ is not connected.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology: $10$