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?

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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?

Mathematics
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{x: \frac{1}{2}\ < |x| < 1}
First, let us talk about what an open set will look like.
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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Other answers:

Every open set in \(\mathbb{R}\) can be written as the union of intervals of the form \((a, b)\).
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?
please wait let me think
yes , it can be written like that
Show me.
\[(-1,1/2) \cup(1/2,1)\]
so that the union =(-1,1)
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 \([-1,1]\)
yes but please explain this \[[−1,1]∩[(−1,−1/2)∪(1/2/1)] \]
i know that [−1,1]∩[(−1,−1/2)∪(1/2,1)] is open in R
but i know that [-1,1] is close in R.
and in the subspace. because it can be written in the form of [-1,1] intersected with an open set in R
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.
I really think if you ask about one of the other options, this one will make more sense...
ok but do you have any examples to make me understand more?
Didn't they give you more options? like I think \[\{x\mid \frac{1}{2}\le |x| < 1\}\] Was this not an option?
and \(\{x\mid \frac{1}{2}< |x| \le 1\}\) and \(\{x\mid \frac{1}{2}\le |x| \le 1\}\)
You here man? These are yes or no questions :)
YES THEY DID.
NETWORK IS TOOOOOO POOR
OK, NOW it means that the option that is close is \[{{x∣1/2≤|x|≤1} } \] is the close set. ?
now, with same options, which of them are open in the standard topology on R and which of them are close?
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
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.
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 ?
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.
Which one would you like to look at?
\[{x∣1/2<|x|≤1} \] this one first, is it open or close in R . i guess it is neither
Try and write that set as the union of open intervals.
will it be \[(-1,-\frac{ 1 }{ 2})\cup(\frac{ 1 }{ 2 },1]\]
am i right sir?
which book or site can i lean how to do this sir?
i thick it can not be written
\([-1,-\frac{ 1 }{ 2})\cup(\frac{ 1 }{ 2 },1]\)
The book I gave you covers this in detail.
Answer this please \([(-2,\dfrac{-1}{2})\cup(\dfrac{1}{2},2)]\cap[-1,1]= \ ?\)
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
Just answer the last question I asked
the answer to that question you asked is (-1,1)
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?
i have not studied sub base sir
i am confuse
I have said about 5 times the answer to that question. This lets me know that you are not really studying anything I say.
at first i thought the answer to that your question is empty set but i was not sure
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.
I am not trying to be rude, but it is a waste of time.
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.
thank you sir
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
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
we are talking about the first question.
or at least I am.
ohh. i thought we have answered the first question
Ok Close this and ask the new question :)

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