On continuous one-to-one functions on sets of real numbers. (English) Zbl 0943.26006

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 \(\mathbb R\)?


26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54C05 Continuous maps