Cantor Space is not Locally Connected

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Theorem

Let $T = \left({\mathcal C, \tau_d}\right)$ be the Cantor space.


Then $T$ is not locally connected.


Proof

Let $\mathcal B$ be a basis of $T$.

Let $A \in \mathcal B$.

By definition of $\mathcal B$, $A$ is an open set of $T$.

But the Cantor Space is Totally Separated.

Therefore $A$ is not a connected set.

Hence the result from definition of a locally connected space.

$\blacksquare$


Sources