Let
R
, the real line be endowned with the discrete topology. Which of the following subsets of
R
is dense in
R

Question

Let
R
, the real line be endowned with the discrete topology. Which of the following subsets of
R
is dense in
R

anonymous · Answer

Q
Ritself
Qc
All singletons

steve816 · Answer

Wow, this is some advanced stuff. Just wondering, what branch of mathematics is this?

anonymous · Answer

topology

steve816 · Answer

Nice, wish there was topology class in high school, did you learn about the banach-tarski paradox?

anonymous · Answer

no

steve816 · Answer

Well, it's pretty interesting. I don't think anyone in openstudy knows anything about topology though.

anonymous · Answer

someone does

steve816 · Answer

At least not at 3 AM in the morning lol

anonymous · Answer

yes

steve816 · Answer

So... how hard is topology?

anonymous · Answer

in a way very

zzr0ck3r · Answer

All sets are open in the discrete topology.

Consider any any set $U\ne X$. Then $U^C$ is an open set disjoint from $U$.

This shows that the only dense set is the whole space.

zzr0ck3r · Answer

$U$ is dense means that for any $x$ in the space $X=\mathbb{R} _D$, either $x$ is in $U$ or $x$ is a limit point of $U$. So we take an element $x$ that is not in $U$ and see if it is a limit point of $U$. If $x$ is a limit point of $U$ then every open neighborhood around $x$ intersects $U$. But since every set is open, due to the topology, we have that $U^C$ itself is open and contains $x$ (remember $x\notin U$) and of course $U\cap U^C=\emptyset$. So $U$ contains all of its limit points. Since $U$ was not all of $\mathbb{R}_D$ we have that the only dense set is the entire space.

zzr0ck3r · Answer

Clearly the entire space is dense because it contains all of its own points!

zzr0ck3r · Answer

@steve816 it really is not, and it is a fun new (new to me...)way of looking at things :)

anonymous · Answer

so, which option is correct

anonymous · Answer

is it R?

zzr0ck3r · Answer

It should be very clear. Read it and tell me if you don't understand.

zzr0ck3r · Answer

I want you to at least be sure yourself. I did not write all of that out so that you could ask which option it is, click a button, and move on.

anonymous · Answer

ok, from what you said, i its R

anonymous · Answer

@zzr0ck3r