On continuous one-to-one functions on sets of real numbers (Q1978888)

Two topological spaces \(X\) and \(Y\) are called a special pair if there exist continuous one-to-one mappings of~\(X\) onto~\(Y\) and of~\(Y\) onto~\(X\), but \(X\)~and~\(Y\) are not homeomorphic. Some examples of special pairs are constructed where both members are subspaces of the real line~\(\mathbb R\): (1) \({X_1=E_1}\), \({X_2=E_1\setminus \{p\}}\) where \(X_1\) is countable, \(p\in E_1\); (2) \(X_2\), \(Y_2\) being closed (uncountable) subspaces of~\(\mathbb R\); (3) \(X_3=E_3\), \(Y_3=E_3\setminus \{p\}\) where \(E_3\) is closed, \(p\in \mathbb R\); (4) \(X_4=\bigcup _{i=1}^\infty J_i\), \(Y_4=\bigcup _{i=2}^\infty J_i\) where \(J_1,J_2,\dots \) is a sequence of mutually disjoint compact intervals. Two questions are posed: \((1')\) Can \(E_1\) be a closed countable subset of~\(\mathbb R\)? \((2')\) Can \(X_2\) and \(Y_2\) be closed countable subsets of~\(\mathbb R\)?