Mutual aposyndesis of symmetric products (Q2701852)

Let \(X\) be a continuum. Then \(X\) is aposyndetic provided that for all distinct \(x,y\in X\) there is a subcontiuum \(K\) of \(X\) such that \(K\) is a neighborhood of \(x\) and \(y\not\in K\). In addition, \(X\) is mutually aposyndetic if for any two different points \(x,y\in X\) there are disjoint subcontinua \(K\) and \(L\) in \(X\) such that \(x\) is in the interior of \(K\) and \(y\) is in the interior of \(L\). It is known that the product of three nondegenerate continua is mutually aposyndetic. Responding to a problem posed by Illanes, the author proves that for every continuum \(X\) its symmetric product \(F_n(X)\), \(n\geq 3\), is mutually aposyndetic.