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locally path-connected space

topological space with a base of path-connected sets

en-US/ˈloʊkəli ˈpæθ - kəˈnɛktɪd ˈspeɪs/
en-GB/ˈləʊkəlɪ pɑːθ - kəˈnɛktɪd ˈspeɪs/

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locally connected space