My Math Forum Doubts on topology

 Topology Topology Math Forum

February 27th, 2017, 04:24 PM   #11
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 Originally Posted by Maschke 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.
I am actually going through some series of videos to understand the concept of topology. I will work out the proof. Could you please provide me definitions for open set and interior points if possible ðŸ˜Š

February 27th, 2017, 04:57 PM   #12
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 Originally Posted by Lalitha183 I am actually going through some series of videos to understand the concept of topology. I will work out the proof. Could you please provide me definitions for open set and interior points if possible ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½

You have already posted images of your notes with these definitions, although we are still trying to work through the slight inaccuracy in your definition of an interior point.

But why do you need me to repeat these to you? You have already posted them. You should write down the exact definitions in the videos and then ask questions about them.

It's hard to learn this material only from videos. You would benefit from a book. I'm sure you can find something online. I'm not familiar with the standard texts on general topology that are currently regarded as good. You can certainly find PDF copies of topology texts online.

Have you already studied real analysis? That's a very helpful prerequisite for topology.

Last edited by Maschke; February 27th, 2017 at 05:01 PM.

February 27th, 2017, 04:59 PM   #13
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 Originally Posted by Maschke You have already posted images of your notes with these definitions, although we are still trying to work through the slight inaccuracy in your definition of an interior point. But why do you need me to repeat these to you? You have already posted them. Are you in a class? Self-studying from a book or websites? I'm trying to put your questions in context.
Im little confused with them. As you have mentioned that slight difference might lead to misunderstand the concept. Anyway I will ensure its definition by its application. I need some more time for this.

Ah no! Im learning it through web lectures on youtube and referring some books. I am actually preparing for an entrance test which includes this topic which I never learned in my graduation.

Last edited by Lalitha183; February 27th, 2017 at 05:02 PM.

February 27th, 2017, 05:18 PM   #14
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 Originally Posted by Lalitha183 Im little confused with them. As you have mentioned that slight difference might lead to misunderstand the concept. Anyway I will ensure its definition by its application. I need some more time for this. Ah no! Im learning it through web lectures on youtube and referring some books. I am actually preparing for an entrance test which includes this topic which I never learned in my graduation.
Ok here is the general idea. Think about the real numbers $\mathbb R$ along with the usual absolute value function $\lvert \cdot \rvert: \mathbb R \to \mathbb R_{\geq 0}$ sending the real number $x$ to its absolute value $\lvert x \rvert$.

$\lvert \cdot \rvert$ induces a distance function, or metric on the real numbers given by $d(x,y) = \lvert x - y \rvert$. Using this metric we can define open intervals like $(a,b)$, and by taking unions of open intervals we can get open sets. [That's not yet a rigorous definition, I'm speaking casually here].

Now you can (and should!) prove that the union of open sets is open; that a finite intersection of open sets is open; that the empty set is open; and that the set of real numbers is open (as a subset of the real numbers).

If you have seen this material before, please let me know. If not, please let me know. You can learn topology either way, but it's helpful to me to know whether you've seen this material in the context of the real numbers before. That's typically done in a class called real analysis.

Now the idea with topology is to abstract the properties of open sets in the real numbers to more general situations. So we make the following definition:

A set $X$ along with a collection $\mathscr T$ of subsets of $X$ is called a topological space if $\mathscr T$ is closed (sometimes called stable) under arbitrary unions and finite intersections, and the empty set and $X$ are in $\mathscr T$.

So the conversation now branches. If you have pretty much seen these ideas in the context of the real numbers, then that's one conversation.

But if what I've written is new to you, we can still learn topology but we have to review the situation in the real numbers first.

Is this helpful? In theory if we were perfectly logical beings we could just take the rules of topology and crank out all the theorems. But in real life we learn the general from the specific, and we get all our basic topological intuitions from the real numbers. So we have to make sure we understand the topology of the real numbers first.

February 27th, 2017, 05:44 PM   #15
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 Originally Posted by Maschke Ok here is the general idea. Think about the real numbers $\mathbb R$ along with the usual absolute value function $\lvert \cdot \rvert: \mathbb R \to \mathbb R_{\geq 0}$ sending the real number $x$ to its absolute value $\lvert x \rvert$. $\lvert \cdot \rvert$ induces a distance function, or metric on the real numbers given by $d(x,y) = \lvert x - y \rvert$. Using this metric we can define open intervals like $(a,b)$, and by taking unions of open intervals we can get open sets. [That's not yet a rigorous definition, I'm speaking casually here]. Now you can (and should!) prove that the union of open sets is open; that a finite intersection of open sets is open; that the empty set is open; and that the set of real numbers is open (as a subset of the real numbers). If you have seen this material before, please let me know. If not, please let me know. You can learn topology either way, but it's helpful to me to know whether you've seen this material in the context of the real numbers before. That's typically done in a class called real analysis. Now the idea with topology is to abstract the properties of open sets in the real numbers to more general situations. So we make the following definition: A set $X$ along with a collection $\mathscr T$ of subsets of $X$ is called a topological space if $\mathscr T$ is closed (sometimes called stable) under arbitrary unions and finite intersections, and the empty set and $X$ are in $\mathscr T$. So the conversation now branches. If you have pretty much seen these ideas in the context of the real numbers, then that's one conversation. But if what I've written is new to you, we can still learn topology but we have to review the situation in the real numbers first. Is this helpful? In theory if we were perfectly logical beings we could just take the rules of topology and crank out all the theorems. But in real life we learn the general from the specific, and we get all our basic topological intuitions from the real numbers. So we have to make sure we understand the topology of the real numbers first.
I didn't see it w.r.t $R$
I understood the definitions of topology but I still didn't went through number systems to see its applicability.

February 27th, 2017, 05:53 PM   #16
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Quote:
 Originally Posted by Maschke Ok here is the general idea. Think about the real numbers $\mathbb R$ along with the usual absolute value function $\lvert \cdot \rvert: \mathbb R \to \mathbb R_{\geq 0}$ sending the real number $x$ to its absolute value $\lvert x \rvert$. $\lvert \cdot \rvert$ induces a distance function, or metric on the real numbers given by $d(x,y) = \lvert x - y \rvert$. Using this metric we can define open intervals like $(a,b)$, and by taking unions of open intervals we can get open sets. [That's not yet a rigorous definition, I'm speaking casually here]. Now you can (and should!) prove that the union of open sets is open; that a finite intersection of open sets is open; that the empty set is open; and that the set of real numbers is open (as a subset of the real numbers). If you have seen this material before, please let me know. If not, please let me know. You can learn topology either way, but it's helpful to me to know whether you've seen this material in the context of the real numbers before. That's typically done in a class called real analysis. Now the idea with topology is to abstract the properties of open sets in the real numbers to more general situations. So we make the following definition: A set $X$ along with a collection $\mathscr T$ of subsets of $X$ is called a topological space if $\mathscr T$ is closed (sometimes called stable) under arbitrary unions and finite intersections, and the empty set and $X$ are in $\mathscr T$. So the conversation now branches. If you have pretty much seen these ideas in the context of the real numbers, then that's one conversation. But if what I've written is new to you, we can still learn topology but we have to review the situation in the real numbers first. Is this helpful? In theory if we were perfectly logical beings we could just take the rules of topology and crank out all the theorems. But in real life we learn the general from the specific, and we get all our basic topological intuitions from the real numbers. So we have to make sure we understand the topology of the real numbers first.
Please to through the image. I guess that would be proving A as an Open set ?

February 27th, 2017, 05:56 PM   #17
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Quote:
 Originally Posted by Lalitha183 Please to through the image. I guess that would be proving A as an Open set ?
Sorry i have included the image now.
Attached Images
 20170228_082114.jpg (90.0 KB, 3 views)

February 27th, 2017, 05:57 PM   #18
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Quote:
 Originally Posted by Lalitha183 I didn't see it w.r.t $R$ I understood the definitions of topology but I still didn't went through number systems to see its applicability.
Have you seen the $\epsilon$-$\delta$ definition of continuity in calculus? The $\epsilon$-$N$ definition of the limit of a function? Can you say what math courses you've taken?

Topology will difficult if you haven't seen rigorous proofs involving continuity.

I'm not saying we can't do this. After all topology can in theory be completely self-contained and based only on basic set theory. It's just that if we do that we'll have to spend time studying examples in the real numbers.

How much formal math have you seen?

Last edited by Maschke; February 27th, 2017 at 06:02 PM.

February 27th, 2017, 06:01 PM   #19
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Quote:
 Originally Posted by Lalitha183 Sorry i have included the image now.
I'm afraid I don't see a mathematical argument or exposition there. You start to define a topological space then it gets very confused. Also what is $A^0$, you did not define this notation. Then your normally clear and clean handwriting started going off at an angle and getting messy. I know when I do that it means I'm getting confused! But I didn't see an exposition I could follow.

There is a mismatch between what you know and what you are seeking to learn.

Have you taken a proof-based math course before?

I just need to understand your math experience so I can begin explaining topology at a level that will make sense.

February 27th, 2017, 06:06 PM   #20
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Quote:
 Originally Posted by Maschke I'm afraid I don't see a mathematical argument or exposition there. You start to define a topological space then it gets very confused. Also what is $A^0$, you did not define this notation. Then your normally clear and clean handwriting started going off at an angle and getting messy. I know when I do that it means I'm getting confused! But I didn't see an exposition I could follow. There is a mismatch between what you know and what you are seeking to learn. Have you taken a proof-based math course before? I just need to understand your math experience so I can begin explaining topology at a level that will make sense.
I have taken courses on Algebra, Real analysis and vectors, group theory ring theory.
You mean proof writing theorems or a course which teaches how to write a proof?
I have been into proofs and theorems whole year in 2nd year of my graduation and havent taken any math course that teaches how to write a proof.

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