Theorem

A componentof a locally connected spaceis open.

Proof

Letbe a component of a locally connected spaceand let

Sinceis locally connected, a belongs to at least one open connected set

and since eachis open, so is

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Theorem

A componentof a locally connected spaceis open.

Proof

Letbe a component of a locally connected spaceand let

Sinceis locally connected, a belongs to at least one open connected set

and since eachis open, so is