Non-Trivial Discrete Space is not Connected

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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.

$\blacksquare$


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