Question

R^n is open and closed set.. how??
please help

Answer 1

R^n is an open set, right?
That much is clear?

Answer 2

In that given any element x of R^n, there exists a neighbourhood of x contained entirely in R^n

Answer 3

Remember... if a set is not open, it doesn't necessarily mean it's closed.
In a similar notion... open does not necessarily mean not closed ;)

Answer 4

you may need to bring this question to mathstackexchange.com
this may also help
http://www.tricki.org/article/To_prove_that_a_set_is_open_or_closed_use_basic_theorems_rather_than_direct_arguments

Answer 5

very interesting question, I thought \(\mathbb R^n\) was closed

Answer 6

It is. It is also open.

Answer 7

A closed set is simply a set whose complement is open, right? :)

Answer 8

hmmm..
\[\huge (\mathbb{R}^n)^C=\emptyset\]

Answer 9

And proving that the null set is open is tricky :D or not

Answer 10

Suppose the null set is not open. Then there exists an element in the null set such that it has a neighbourhood not entirely contained in the null set.
And that's a contradiction in itself.

"There exists an element in the null set..." is a contradiction.
Hence, trivially, the null set is open, hence R^n is closed

Answer 11

I follow your argument, but don't know enough set theory to be sure you can use the null set as an example. I don't see why not though.

Answer 12

So...
simply put... the metric space is R^n and
\[\huge \mathbb{R}^n\cap (\mathbb{R}^n)'=\emptyset\]

Answer 13

I will have to take your word for that, bro. Like I said, I know very little set theory.
Nice job!

Answer 14

Thanks :)
Now it's off to bed for me :D

Answer 15

g'night!