= Connected locally path-connected spaces are path-connected
In a <locally path-connected space>, every <path component> is <open>: a <path-connected> open neighbourhood of a point lies within that point's <path component>. The complement of a <path component> is a union of the other <open> <path components>, hence is also <open>. In a nonempty <connected space>, a nonempty subset that is both <open> and <closed> must be the whole space. Thus a <connected space> that is <locally path-connected> is <path-connected>.
Back to article page
