A topological space has local connectedness when every point has a neighborhood basis of connected open sets. In particular, a planar domain has local connectedness, but its domain boundary need not be. Local connectedness of the boundary is the condition governing a continuous extension of a conformal bijection to the parametrizing circle.
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In topology, a space is said to be **locally connected** at a point if every neighborhood of that point contains a connected neighborhood of that point. More formally, a topological space \(X\) is said to be **locally connected** if for every point \(x \in X\) and every neighborhood \(U\) of \(x\), there exists a connected neighborhood \(V\) of \(x\) such that \(V \subseteq U\).