Let $A$ be a subset of $\Bbb{R}$ such that the following $7$ sets are all different

Question

I am suppose to come up with an example of an subset $A$ such that the sets

$ A$

$int(A)$

$cl(A)$

$ cl(int(A))$

$ int(cl(A))$

$int(cl(int(A)))$

$ cl(int(cl(A)))$

are all different. I am not really sure how to come up with example like that so any hint or idea is great.

If you can find an example of an $A$ that gives a difference for each pair of descriptions, then you can squash each of those 21 examples into finite space by applying arctan to it, and then translate them away from each other and deliver them all as a single subset ... — hmakholm left over Monica, Jan 23 '16 at 15:51
Possible duplicate of Playing with closure and interior to get the maximum number of sets — Najib Idrissi, Jan 23 '16 at 16:17

Balloon · Accepted Answer · 2016-01-23T16:11:19.000

5

Let $A=[0,1]\cup(2,3)\cup([4,5]\cap\mathbb{Q})\cup\{6\}\cup(7,8)\cup(8,9).$ Then note that :

edited Jan 23 '16 at 16:11

answered Jan 23 '16 at 16:03

Balloon

I edited that ! – Balloon Jan 23 '16 at 16:12
2

I think also just $A = [(0,1) \cap \mathbb{Q}] \cup (1,2) \cup (2,3) \cup {4}$ works – Dan Simon Jan 23 '16 at 16:14
@Dan Simon : Yes you are right, and it is more elegant. I tried to add different kinds of sets step-by-step, but it seems there are too numerous now ! ^^ – Balloon Jan 23 '16 at 16:18

1 Answers1