Motivating topological proofs
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Hi I'm working on Rudin's real analysis textbook and I motivate the proofs about topology using $R$ $R^2$ and $R^3$ in my mind but I know he proves it for $R^n$. Even though I know if it works for the latter it should work the former, I am wondering if I should totally forget about the geometric intuition and solely work with logic and definitions for proving things more accurately in my mind as well. Or does he also motivate them with the basic metric spaces and just generalizes it to $R^n$?
real-analysis general-topology real-numbers
asked Dec 27 '18 at 2:12
Kaan YolseverKaan Yolsever
1309
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add a comment |
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Hi I'm working on Rudin's real analysis textbook and I motivate the proofs about topology using $R$ $R^2$ and $R^3$ in my mind but I know he proves it for $R^n$. Even though I know if it works for the latter it should work the former, I am wondering if I should totally forget about the geometric intuition and solely work with logic and definitions for proving things more accurately in my mind as well. Or does he also motivate them with the basic metric spaces and just generalizes it to $R^n$?
real-analysis general-topology real-numbers
asked Dec 27 '18 at 2:12
Kaan YolseverKaan Yolsever
1309
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2
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You might be interested in reading the preface to Visual Complex Analysis by Needham. "Lockhart's Lament" is also good.
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– littleO
Dec 27 '18 at 2:47
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Many results about $Bbb R^n$ hold for (1) all metric spaces, or (2) all complete metric spaces, or (3) all metric spaces in which closed bounded subsets are compact. But I endorse the A from Robert Israel. In topology I always try to think in pictures.
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– DanielWainfleet
Dec 27 '18 at 18:39
add a comment |
$begingroup$
Hi I'm working on Rudin's real analysis textbook and I motivate the proofs about topology using $R$ $R^2$ and $R^3$ in my mind but I know he proves it for $R^n$. Even though I know if it works for the latter it should work the former, I am wondering if I should totally forget about the geometric intuition and solely work with logic and definitions for proving things more accurately in my mind as well. Or does he also motivate them with the basic metric spaces and just generalizes it to $R^n$?
real-analysis general-topology real-numbers
asked Dec 27 '18 at 2:12
Kaan YolseverKaan Yolsever
1309
$endgroup$
Hi I'm working on Rudin's real analysis textbook and I motivate the proofs about topology using $R$ $R^2$ and $R^3$ in my mind but I know he proves it for $R^n$. Even though I know if it works for the latter it should work the former, I am wondering if I should totally forget about the geometric intuition and solely work with logic and definitions for proving things more accurately in my mind as well. Or does he also motivate them with the basic metric spaces and just generalizes it to $R^n$?
real-analysis general-topology real-numbers
real-analysis general-topology real-numbers
asked Dec 27 '18 at 2:12
Kaan YolseverKaan Yolsever
1309
asked Dec 27 '18 at 2:12
Kaan YolseverKaan Yolsever
1309
asked Dec 27 '18 at 2:12
Kaan YolseverKaan Yolsever
1309
asked Dec 27 '18 at 2:12
Kaan YolseverKaan Yolsever
1309
asked Dec 27 '18 at 2:12
Kaan YolseverKaan Yolsever
1309
1309
2
$begingroup$
You might be interested in reading the preface to Visual Complex Analysis by Needham. "Lockhart's Lament" is also good.
$endgroup$
– littleO
Dec 27 '18 at 2:47
$begingroup$
Many results about $Bbb R^n$ hold for (1) all metric spaces, or (2) all complete metric spaces, or (3) all metric spaces in which closed bounded subsets are compact. But I endorse the A from Robert Israel. In topology I always try to think in pictures.
$endgroup$
– DanielWainfleet
Dec 27 '18 at 18:39
add a comment |
2
$begingroup$
You might be interested in reading the preface to Visual Complex Analysis by Needham. "Lockhart's Lament" is also good.
$endgroup$
– littleO
Dec 27 '18 at 2:47
$begingroup$
Many results about $Bbb R^n$ hold for (1) all metric spaces, or (2) all complete metric spaces, or (3) all metric spaces in which closed bounded subsets are compact. But I endorse the A from Robert Israel. In topology I always try to think in pictures.
$endgroup$
– DanielWainfleet
Dec 27 '18 at 18:39
2
2
$begingroup$
You might be interested in reading the preface to Visual Complex Analysis by Needham. "Lockhart's Lament" is also good.
$endgroup$
– littleO
Dec 27 '18 at 2:47
$begingroup$
You might be interested in reading the preface to Visual Complex Analysis by Needham. "Lockhart's Lament" is also good.
$endgroup$
– littleO
Dec 27 '18 at 2:47
$begingroup$
Many results about $Bbb R^n$ hold for (1) all metric spaces, or (2) all complete metric spaces, or (3) all metric spaces in which closed bounded subsets are compact. But I endorse the A from Robert Israel. In topology I always try to think in pictures.
$endgroup$
– DanielWainfleet
Dec 27 '18 at 18:39
$begingroup$
Many results about $Bbb R^n$ hold for (1) all metric spaces, or (2) all complete metric spaces, or (3) all metric spaces in which closed bounded subsets are compact. But I endorse the A from Robert Israel. In topology I always try to think in pictures.
$endgroup$
– DanielWainfleet
Dec 27 '18 at 18:39
add a comment |
1 Answer
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You should definitely not forget about geometric intuition. It is a very powerful tool, even in high dimensions. $mathbb R^n$ is much like $mathbb R^2$ or $mathbb R^3$, it just has a few extra dimensions thrown in.
answered Dec 27 '18 at 2:45
Robert IsraelRobert Israel
327k23216470
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$begingroup$
You should definitely not forget about geometric intuition. It is a very powerful tool, even in high dimensions. $mathbb R^n$ is much like $mathbb R^2$ or $mathbb R^3$, it just has a few extra dimensions thrown in.
answered Dec 27 '18 at 2:45
Robert IsraelRobert Israel
327k23216470
$endgroup$
add a comment |
$begingroup$
You should definitely not forget about geometric intuition. It is a very powerful tool, even in high dimensions. $mathbb R^n$ is much like $mathbb R^2$ or $mathbb R^3$, it just has a few extra dimensions thrown in.
answered Dec 27 '18 at 2:45
Robert IsraelRobert Israel
327k23216470
$endgroup$
add a comment |
$begingroup$
You should definitely not forget about geometric intuition. It is a very powerful tool, even in high dimensions. $mathbb R^n$ is much like $mathbb R^2$ or $mathbb R^3$, it just has a few extra dimensions thrown in.
answered Dec 27 '18 at 2:45
Robert IsraelRobert Israel
327k23216470
$endgroup$
You should definitely not forget about geometric intuition. It is a very powerful tool, even in high dimensions. $mathbb R^n$ is much like $mathbb R^2$ or $mathbb R^3$, it just has a few extra dimensions thrown in.
answered Dec 27 '18 at 2:45
Robert IsraelRobert Israel
327k23216470
answered Dec 27 '18 at 2:45
Robert IsraelRobert Israel
327k23216470
answered Dec 27 '18 at 2:45
Robert IsraelRobert Israel
327k23216470
answered Dec 27 '18 at 2:45
Robert IsraelRobert Israel
327k23216470
327k23216470
add a comment |
add a comment |
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$begingroup$
You might be interested in reading the preface to Visual Complex Analysis by Needham. "Lockhart's Lament" is also good.
$endgroup$
– littleO
Dec 27 '18 at 2:47
$begingroup$
Many results about $Bbb R^n$ hold for (1) all metric spaces, or (2) all complete metric spaces, or (3) all metric spaces in which closed bounded subsets are compact. But I endorse the A from Robert Israel. In topology I always try to think in pictures.
$endgroup$
– DanielWainfleet
Dec 27 '18 at 18:39