On some variants of connectedness (Q1271962)

Suppose that a topological space \(X\) has some topological property \(P\) and that \(M\subset X\) is such that \(\text{fr} (M)\) has \(P\), then is it true that \(\text{cl} (M)\) has \(P\)? In the article under review, this question is answered in the affirmative when \(P\in \{\)connected, pathwise connected, locally connected\(\}\) and counterexamples are given to show that there is a negative answer when \(P\in \{\)arcwise connected, Cantor connected\(\}\).