Connectedness and Path connectedness

Local (path) connectedness

A topological space 𝑋 is (path-)connected iff for each 𝑥 ∈𝑋 and every neighbourhood 𝑈 of 𝑋, there exists a (path-)connected open neighbourhood 𝑉 of 𝑥 such that 𝑉 ⊆𝑈. Equivalently, every point has a neighbourhood basis of (path-)connected sets. topology

Local (path) connectedness is neither weaker nor stronger than (path) connectedness, however Locally path connected spaces have identical connected and path-connected components.

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