Non-Trivial Discrete Space is not Connected

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: $10$