Let R be endowed with the usual standard topology. Consider Y = [-1,1] as a subspace of R. Which one of the following sets is closed in Y?

- anonymous

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- anonymous

{x: \frac{1}{2}\ < |x| < 1}

- zzr0ck3r

First, let us talk about what an open set will look like.

- zzr0ck3r

An open set in the subspace, will be of the form \(A\cap [-1,1]\) where \(A\) is open in \(\mathbb{R}\).

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## More answers

- zzr0ck3r

Every open set in \(\mathbb{R}\) can be written as the union of intervals of the form \((a, b)\).

- zzr0ck3r

Can this be written in the form \(A\cap [-1,1]\) where \(A\) is open in \(\mathbb{R}\)?
If so, show me. If not, why not?

- anonymous

please wait let me think

- anonymous

yes , it can be written like that

- zzr0ck3r

Show me.

- anonymous

\[(-1,1/2) \cup(1/2,1)\]

- anonymous

so that the union =(-1,1)

- anonymous

@zzr0ck3r

- zzr0ck3r

I think you mean the following:
Consider \((-1, -1/2)\cup (1/2, 1)\).
Since \((-1, -1/2)\cup (1/2, 1)\) is open in \(\mathbb{R}\) we have that
\([-1,1]\cap[(-1, -1/2)\cup (1/2, 1)]\) is open in the subspace \([-1,1]\)

- anonymous

yes but please explain this
\[[−1,1]∩[(−1,−1/2)∪(1/2/1)] \]

- anonymous

i know that [−1,1]∩[(−1,−1/2)∪(1/2,1)] is open in R

- anonymous

but i know that [-1,1] is close in R.

- zzr0ck3r

and in the subspace. because it can be written in the form of [-1,1] intersected with an open set in R

- zzr0ck3r

This one is sort of trivial, because it is a subset of [-1,1] so it equals the intersection of itself and [-1,1]. Since it was open in R to begin with, it is open in the subspace.

- zzr0ck3r

I really think if you ask about one of the other options, this one will make more sense...

- anonymous

ok but do you have any examples to make me understand more?

- zzr0ck3r

Didn't they give you more options?
like I think
\[\{x\mid \frac{1}{2}\le |x| < 1\}\]
Was this not an option?

- zzr0ck3r

and \(\{x\mid \frac{1}{2}< |x| \le 1\}\)
and \(\{x\mid \frac{1}{2}\le |x| \le 1\}\)

- zzr0ck3r

You here man? These are yes or no questions :)

- anonymous

YES THEY DID.

- anonymous

NETWORK IS TOOOOOO POOR

- anonymous

OK, NOW it means that the option that is close is \[{{x∣1/2≤|x|≤1} } \] is the close set. ?

- anonymous

now, with same options, which of them are open in the standard topology on R and which of them are close?

- anonymous

my guess is that [1/2, 1] is close in R and (1/2,1} is open in R. am i right ?
what then will [1/2, 1) and (1/2,1] be . open or close ?
please explain

- zzr0ck3r

Are you asked if they are open, or closed. A set can be neither open nor closed. A set can be both open and closed.

- anonymous

which of the options are open in r and which are close in R and which are nither open or close and which are both open and close in the options above ?

- zzr0ck3r

It does not help you at all for me just to tell you. You need to go with the definition and figure it out. Then it will always make sense.

- zzr0ck3r

Which one would you like to look at?

- anonymous

\[{x∣1/2<|x|≤1} \] this one first, is it open or close in R . i guess it is neither

- zzr0ck3r

Try and write that set as the union of open intervals.

- anonymous

will it be \[(-1,-\frac{ 1 }{ 2})\cup(\frac{ 1 }{ 2 },1]\]

- anonymous

am i right sir?

- anonymous

which book or site can i lean how to do this sir?

- anonymous

i thick it can not be written

- anonymous

@zzr0ck3r

- zzr0ck3r

\([-1,-\frac{ 1 }{ 2})\cup(\frac{ 1 }{ 2 },1]\)

- zzr0ck3r

The book I gave you covers this in detail.

- zzr0ck3r

Answer this please
\([(-2,\dfrac{-1}{2})\cup(\dfrac{1}{2},2)]\cap[-1,1]= \ ?\)

- anonymous

i have only learnt open sets, close sets, neither open or close , . but i saw an example like that but it only stated the open and the close sets

- zzr0ck3r

Just answer the last question I asked

- anonymous

the answer to that question you asked is (-1,1)

- zzr0ck3r

not at all. You really need to go spend some time learning basic set theory, else there is no way you will learn this.
How do we know a set is open in a subspace topology?

- anonymous

i have not studied sub base sir

- anonymous

i am confuse

- zzr0ck3r

I have said about 5 times the answer to that question. This lets me know that you are not really studying anything I say.

- anonymous

at first i thought the answer to that your question is empty set but i was not sure

- zzr0ck3r

You are just guessing and repeating things.... you are not learning
The fact that you cant write \(\{x\mid \frac{1}{2}<|x|\le 1\}\) in interval notation tells me that you are very far from being ready for this.

- zzr0ck3r

I am not trying to be rude, but it is a waste of time.

- zzr0ck3r

It took me months to learn what you are trying to brush over in 20 mins, it will not work, you will not pass the test with this method.
I gave you a book, and it has problems in it. You should read the book and do the problems, that is the ONLY way to learn math.
You may ask me any question you want, but I tell you now that you are wasting your time.

- anonymous

thank you sir

- zzr0ck3r

If you want to keep on with this question, then scroll up and find out where I answerd the following question:
How do we know a set is open in a subspace topology?
Find the answer and explain what I said. Then we will continue

- anonymous

i will do that. let me try to learn that, only that i am confuse because the first question was on [-1,1] but now it said which of the options is open in R

- zzr0ck3r

we are talking about the first question.

- zzr0ck3r

or at least I am.

- anonymous

ohh. i thought we have answered the first question

- zzr0ck3r

Ok Close this and ask the new question :)

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