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X IS LOCALLY (PATH) CONNECTED IFF FOR EVERY OPEN SET U IN X EACH (PATH) COMPONENT OF U IS OPEN IN X
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THEOREM AND PROOF:
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X IS LOCALLY (PATH) CONNECTED IFF FOR EVERY OPEN SET U IN X EACH (PATH) COMPONENT OF U IS OPEN IN X
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IF X IS LOCALLY PATH CONNECTED THEN THE COMPONENTS AND PATH COMPONENTS ARE SAME
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if X is locally connected and f is function from x to y then is locally connected
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