Need a counterexample related to Interior and Closure

Question

Let "cl" denotes closure, "int" denotes interior.I'm looking for a Single example of a subset $A$ of some topological space $X$ where all the following sets are unequal:

$1.$A

$2.$ int(A),

$3.$cl(A),

$4.$ int(cl(A)),

$5.$ Cl(int(cl(A))),

$6.$ cl(int(A)),

$7.$ int(cl(int(A)))

It's clear that such a set $A$ is neither open nor closed.Moreover,in $\mathbb R$ with usual topology It's easy to construct some example using intervals and $\mathbb Q$ where some of the above sets are unequal.But I'm finding it a bit hard to construct a example where all the above $7$ sets are unequal.Any hints/ideas?

Related : http://math.stackexchange.com/questions/186017/what-is-the-smallest-cardinality-of-a-kuratowski-14-set — Tom-Tom, Sep 29 '15 at 19:32

score 1 · Accepted Answer · answered Sep 29 '15 at 15:47

1

HINT: Do it in pieces. Let $A_1=[0,1)\cup(1,2)\cup\{3\}$; then $\operatorname{cl}A_1=[0,2]\cup\{3\}$, $\operatorname{int}\operatorname{cl}A_1=(0,2)$, and $\operatorname{cl}\operatorname{int}\operatorname{cl}A_1=[0,2]$. Moreover, $\operatorname{int}A_1=(0,1)\cup(1,2)$, so we’ve taken care of everything except $\operatorname{cl}\operatorname{int}A$ and $\operatorname{int}\operatorname{cl}\operatorname{int}A$.

Now let $A_2=(-1,4)\setminus A_1=(-1,0)\cup\{1\}\cup[2,3)\cup(3,4)$. Show that $A_2$, $\operatorname{cl}A_2$, $\operatorname{int}A_2$, $\operatorname{cl}\operatorname{int}A_2$, and $\operatorname{int}\operatorname{cl}\operatorname{int}A_2$ are all distinct.

Now what if you had a set $A$ that was like a discrete union of $A_1$ and $A_2$? And how can you make such a set?

answered Sep 29 '15 at 15:47

Brian M. Scott

616,228

Sorry..What do you mean by "discrete union" ? – Dontknowanything Sep 29 '15 at 16:30
@Learner: A space $X$ is the discrete union of spaces $Y$ and $Z$ if it is the disjoint union of open subspaces $Y'$ and $Z'$ such that $Y'$ is homeomorphic to $Y$, and $Z'$ is homeomorphic to $Z$. – Brian M. Scott Sep 29 '15 at 16:45
Thanks,So how do you construct such $A$ here using $A_1$ and $A_2$ ?(Sorry for my trivial questions,I'm new to Topology) – Dontknowanything Sep 29 '15 at 16:50
@Learner: Slide $A_2$ to the right far enough not to interfere with $A_1$ and take the union. – Brian M. Scott Sep 29 '15 at 20:31
@BrianM.Scott Sorry for Replying late.As usual,fantastic idea.Thank a lot.Best Regards, – Arpit Kansal Sep 30 '15 at 03:25
@Arpit: You're welcome! – Brian M. Scott Sep 30 '15 at 14:25

score 0 · Answer 2 · answered Sep 29 '15 at 15:44

0

Try $A = ([0,2] \cap \mathbb{Q}) \cup \{ 3 \} \cup (4,5) \cup (5,6) \subseteq \mathbb{R}$, with the usual topology.

Edit: after looking at the "possible duplicate" above, the comments there show an almost identical solution.

answered Sep 29 '15 at 15:44

phaiakia

557

Need a counterexample related to Interior and Closure

2 Answers2