Path-Connected implies Connected
TODO - if it is not connected, then two disjoint open subsets. Take point from either and draw a path between them. Paths are connected because $[0, 1]$ is connected. But disconnected in subspace topology.
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TODO - if it is not connected, then two disjoint open subsets. Take point from either and draw a path between them. Paths are connected because $[0, 1]$ is connected. But disconnected in subspace topology.