Equivalence of Definitions of Locally Connected Space/Definition 1 implies Definition 2
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Theorem
Let $T = \struct{S, \tau}$ be a topological space.
Let each point of $T$ have a local basis consisting entirely of connected sets in $T$.
Then
- each point of $T$ has a neighborhood basis consisting of connected sets of $T$.
Proof
From Local Basis for Open Sets Implies Neighborhood Basis of Open Sets, it follows directly that:
- each point of $T$ has a neighborhood basis consisting entirely of connected sets in $T$.
$\blacksquare$