Theorem
A component
of a locally connected space
is open.
Proof
Letbe a component of a locally connected spaceand let
Since
is locally connected, a belongs to at least one open connected set
and since each
is 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