On the boundary of fuzzy sets (Q1277864)

A new concept of boundary (here called frontier) \(\text{Fr} (\mu)\) of a fuzzy set \(\mu\) in a fuzzy topological space \(X\) is given. The properties of the frontier are studied. Some of the results are as follows. A fuzzy set \(\mu\) in \(X\) is clopen iff \(\text{Fr }\mu= 0_X\); \(\mu\) is not clopen iff \(\text{Fr }\mu\) and \(\text{Fr} (1_X-\mu)\) are overlapping fuzzy sets. If \(\mu\) is a closed fuzzy set of \(X\), then \(\text{Fr }\mu= \text{Fr Fr }\mu\). If \(Y\) is a fuzzy subspace of \(X\) and \(\mu\) is a fuzzy set in \(X\) then \(\text{Fr} (\mu|_Y)\leq \text{Fr } \mu|_Y\) holds.