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February 26th, 2017, 04:42 PM   #1
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Hello there!
I have a doubt, can anyone please clarify!!!
If a set containing interior points as its elements then Is that set needs to be an open set ?

this is new to me!!
Please someone help!!!

Thanks in advance ๐Ÿ˜Š
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February 26th, 2017, 04:54 PM   #2
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Quote:
Originally Posted by Lalitha183 View Post
Hello there!
I have a doubt, can anyone please clarify!!!
If a set containing interior points as its elements then Is that set needs to be an open set ?
tl;dr: Yes, if you mean all points not just some. Your wording needs to be more clear.

That said, some general comments.

If a set contains some interior points, it might still not be an open set. For example if $I = [0,1]$ is the closed unit interval, the point $\frac{1}{2}$ is an interior point, since I can find a neighborhood of $\frac{1}{2}$ entirely contained in $I$.

What you are asking is if ALL of the points of some set are interior points, then it's an open set. Right?

When learning this material it's very important to learn to express your mathematical ideas with precision.

Now the answer to that question depends on the exact definition of an open set, and an interior point. If you write down exactly what is an open set and exactly what is an interior point and exactly what it is you're trying to prove, you'll be able to work out a proof. That's the first step in all these problems, to be very mathematically precise and clear.

I did not read your attachment in much detail, except to note that you have a very clear handwriting and mathematical style. So I'm sure you can transfer this to the message board format.

Toward that end I'd also point you to the use of $\LaTeX$ markup in the forum. You can hit the Quote button on my post to see how it's done. This will serve you greatly in your mathematical career. All math papers are written in $\LaTeX$ these days and if you write up your homework using it, your TA will be endlessly grateful.

I hope you don't mind that I gave you general advice rather than answering your specific question. At the level you're studying math, the advice I gave will help you a lot more than just telling you that yes, a set is open if and only if ALL its points are interior points. That's the answer to the question. You should see if you can prove this directly from the definitions, whichever definitions you've been given. There's more than one way to develop this material so you always have to define your terms.

Last edited by Maschke; February 26th, 2017 at 05:02 PM.
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February 26th, 2017, 05:55 PM   #3
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Quote:
Originally Posted by Maschke View Post
tl;dr: Yes, if you mean all points not just some. Your wording needs to be more clear.

That said, some general comments.

If a set contains some interior points, it might still not be an open set. For example if $I = [0,1]$ is the closed unit interval, the point $\frac{1}{2}$ is an interior point, since I can find a neighborhood of $\frac{1}{2}$ entirely contained in $I$.

What you are asking is if ALL of the points of some set are interior points, then it's an open set. Right?

When learning this material it's very important to learn to express your mathematical ideas with precision.

Now the answer to that question depends on the exact definition of an open set, and an interior point. If you write down exactly what is an open set and exactly what is an interior point and exactly what it is you're trying to prove, you'll be able to work out a proof. That's the first step in all these problems, to be very mathematically precise and clear.

I did not read your attachment in much detail, except to note that you have a very clear handwriting and mathematical style. So I'm sure you can transfer this to the message board format.

Toward that end I'd also point you to the use of $\LaTeX$ markup in the forum. You can hit the Quote button on my post to see how it's done. This will serve you greatly in your mathematical career. All math papers are written in $\LaTeX$ these days and if you write up your homework using it, your TA will be endlessly grateful.

I hope you don't mind that I gave you general advice rather than answering your specific question. At the level you're studying math, the advice I gave will help you a lot more than just telling you that yes, a set is open if and only if ALL its points are interior points. That's the answer to the question. You should see if you can prove this directly from the definitions, whichever definitions you've been given. There's more than one way to develop this material so you always have to define your terms.
Thanks for your advice. I will ensure them from now onwards๐Ÿ˜Š
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February 26th, 2017, 06:08 PM   #4
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Ok now I got interested in working through a clear example of how this problem should be done. So the first thing I did is to look at your picture again, to see if I could figure out how you define an interior point. And I found a little problem.

Quoting from your paper:

Let $(X, \mathscr T)$ be a topological space and $A \subseteq X$. A point $x \in A$ is called an interior point if $\exists U \in \mathscr T$ such that $x \in U$ and $U \subseteq A$.

Note that I didn't bother to mention that $U$ is an open set, since if $U \in \mathscr T$ then it's open by definition.

Now I claim this is an inaccurate definition. According to this definition, every subset of any topological space is an open set. Take $I = [0,1]$ in the reals, which we already know is not an open set. Consider the point $1$. The entire set of reals $\mathbb R$ is an open set (why?) and clearly $1 \in \mathbb R \subseteq \mathbb R$.

So by your definition, $1$ is an interior point of $I$ when in fact it is not.

Do you see what went wrong? We have to require that $U \subsetneq X$, in other words that $U$ is a proper subset of $X$.

Markup note: If you Quote my post, even if you don't reply to it, you can see how I marked up my text to display the math symbols. You don't have to learn this all at once, it takes a while. But if you're planning to study more math, it's a great skill to pick up.

Last edited by Maschke; February 26th, 2017 at 06:20 PM.
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February 26th, 2017, 07:07 PM   #5
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Quote:
Originally Posted by Maschke View Post
Ok now I got interested in working through a clear example of how this problem should be done. So the first thing I did is to look at your picture again, to see if I could figure out how you define an interior point. And I found a little problem.

Quoting from your paper:

Let $(X, \mathscr T)$ be a topological space and $A \subseteq X$. A point $x \in A$ is called an interior point if $\exists U \in \mathscr T$ such that $x \in U$ and $U \subseteq A$.

Note that I didn't bother to mention that $U$ is an open set, since if $U \in \mathscr T$ then it's open by definition.

Now I claim this is an inaccurate definition. According to this definition, every subset of any topological space is an open set. Take $I = [0,1]$ in the reals, which we already know is not an open set. Consider the point $1$. The entire set of reals $\mathbb R$ is an open set (why?) and clearly $1 \in \mathbb R \subseteq \mathbb R$.

So by your definition, $1$ is an interior point of $I$ when in fact it is not.

Do you see what went wrong? We have to require that $U \subsetneq X$, in other words that $U$ is a proper subset of $X$.

Markup note: If you Quote my post, even if you don't reply to it, you can see how I marked up my text to display the math symbols. You don't have to learn this all at once, it takes a while. But if you're planning to study more math, it's a great skill to pick up.


I actually doubted on the notation of the same. please look into the image attached. (written with pencil)
Sorry for the inconvinience as I need more time to write in the markup language.
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February 26th, 2017, 07:14 PM   #6
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Quote:
Originally Posted by Lalitha183 View Post
I actually doubted on the notation of the same. please look into the image attached. (written with pencil)
Sorry for the inconvinience as I need more time to write in the markup language.
I'm afraid the image has nothing to do with what we're talking about. You've given the definition of a topology and mentioned the indiscrete topology. Did you post the wrong image?
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February 26th, 2017, 07:18 PM   #7
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Quote:
Originally Posted by Maschke View Post
I'm afraid the image has nothing to do with what we're talking about. You've given the definition of a topology and mentioned the indiscrete topology. Did you post the wrong image?
Ah no!
You have mentioned in the previous reply that U should be a proper subset of X.
I have written that in mathematical way with pencil in the image attached. I wish that is correct ?!
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February 26th, 2017, 07:30 PM   #8
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Quote:
Originally Posted by Lalitha183 View Post
Ah no!
You have mentioned in the previous reply that U should be a proper subset of X.
I have written that in mathematical way with pencil in the image attached. I wish that is correct ?!
I'm sorry, I see nothing there that has anything to do with the proper definition of an interior point.

Your definition of a topological space is correct (though a little disorganized for my taste) but we are talking about the definition of an interior point.

Last edited by Maschke; February 26th, 2017 at 07:33 PM.
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February 26th, 2017, 08:06 PM   #9
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Quote:
Originally Posted by Maschke View Post
I'm sorry, I see nothing there that has anything to do with the proper definition of an interior point.

Your definition of a topological space is correct (though a little disorganized for my taste) but we are talking about the definition of an interior point.
Oh sorry I was confused with interior points and open set.
Anyway thanks I will rectify the definition ๐Ÿ˜Š
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February 27th, 2017, 12:51 PM   #10
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Standing by. Ok if you've lost interest or figured it out on your own. Just in case you're waiting for me to respond, I'd be glad to walk through a rigorous proof that a set is open if and only if all its points are interior points.
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