Consider sets in R^2 with the usual topology. Is the closure of a set connected by paths connected by paths? does the reverse hold? (That is, if the closure of the set is path-connected, is the set path-connected?)   Please argue your answer and give definitions if necessary. Thank you

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Consider sets in R^2 with the usual topology. Is the closure of a set connected by paths connected by paths? does the reverse hold? (That is, if the closure of the set is path-connected, is the set path-connected?)
 
Please argue your answer and give definitions if necessary. Thank you
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