Example of non G-delta set

Question

An open set is clearly a $G_{\delta}$ set. A closed interval $[a,b]$ is a $G_{\delta}$ set as an intersection of the open intervals $(a-\frac1n,b+\frac1n)$ for all positive integers $n$. What is an example of a set that's not $G_{\delta}$?

score 4 · Accepted Answer · edited Apr 13 '17 at 12:21

4

The set $\Bbb Q$ of rational numbers is not a $G_\delta$ set; you can find several proofs here.

edited Apr 13 '17 at 12:21

Community

1

answered Jun 29 '13 at 00:17

Brian M. Scott

631,399

score 1 · Answer 2 · answered Jun 29 '13 at 00:17

1

It can be shown using something called the Baire Category Theorem that the set of all rational numbers is not a $G_{\delta}$ set.

answered Jun 29 '13 at 00:17

Zarrax

46,591

Example of non G-delta set

2 Answers2