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].(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Construct uncountably dense and holes everywhere

Loading...

Similar Threads - Construct uncountably dense | Date |
---|---|

I Generalization of measure theory to uncountable unions | Sep 19, 2017 |

I Constructing dimensions out of a graph structure? | Oct 3, 2016 |

Constructing atlas | Mar 10, 2014 |

Constructing the real number system by Dedekind cuts? | Dec 26, 2013 |

Construct an explicit isomorphism | Dec 24, 2013 |

**Physics Forums - The Fusion of Science and Community**