- #1
jostpuur
- 2,116
- 19
I want to define a set [itex]X\subset [0,1][/itex], without using the axiom of choice, with the following property: Both [itex][a,b]\cap X[/itex] and [itex][a,b]\backslash X[/itex] are uncountable for all [itex][a,b]\subset [0,1][/itex] where [itex]a<b[/itex].
I don't know how to define [itex]X[/itex] so that [itex][a,b]\cap X[/itex] would always be uncountable without [itex]X[/itex] containing some interval. But if it contains some interval, then [itex][a,b]\backslash X[/itex] will be empty sometimes.
I don't know how to define [itex]X[/itex] so that [itex][a,b]\cap X[/itex] would always be uncountable without [itex]X[/itex] containing some interval. But if it contains some interval, then [itex][a,b]\backslash X[/itex] will be empty sometimes.