# Non-Trivial Discrete Space is not Connected

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## Contents

## Theorem

Let $T = \left({S, \tau}\right)$ 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 $T = \left({S, \tau}\right)$ be a non-trivial discrete space.

Let $a \in S$.

Let $A = \left\{{a}\right\}$ and $B = \complement_S \left({\left\{{a}\right\}}\right)$, where $\complement_S \left({\left\{{a}\right\}}\right)$ is the complement of $A$ in $S$.

As $T$ is not trivial, $B \ne \varnothing$.

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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 1 - 3: \ 10$