An irrational problem
Volume 175 / 2002
Fundamenta Mathematicae 175 (2002), 259-269
MSC: Primary 03C62, 03E35, 54A35, 54B99; Secondary 03E15, 03E55, 54F65, 54G20, 54H05.
DOI: 10.4064/fm175-3-3
Abstract
Given a topological space $\langle X , {\cal T}\rangle \in M$, an elementary submodel of set theory, we define $X_M$ to be $X\cap M$ with topology generated by $\{ U \cap M : U \in {\cal T} \cap M \}$. Suppose $X_M$ is homeomorphic to the irrationals; must $X=X_M$? We have partial results. We also answer a question of Gruenhage by showing that if $X_M$ is homeomorphic to the “Long Cantor Set”, then $X= X_M$.
