Mixing
Interior, exterior and boundary points in $mathbb{R}^n$
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I'm currently taking a class on multivariable calculus. The textbook begins by defining interior, exterior and boundary points in $mathbb{R}^n$ and states that, given a set $Msubsetmathbb{R}^n$, every point $mathbf{a}inmathbb{R}^n$ is either an interior, exterior or boundary point of $M$.
I recognize this must be true for $n = 1, 2, 3$ for geometric reasons but fail to see why it holds in $mathbb{R}^n$ where $n>3$. I tried proving it using the definitions of interior, exterior and boundary points in $mathbb{R}^n$ but got nowhere.
multivariable-calculus
asked Jan 22 at 13:12
S. DoeS. Doe
62
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show 2 more comments
$begingroup$
I'm currently taking a class on multivariable calculus. The textbook begins by defining interior, exterior and boundary points in $mathbb{R}^n$ and states that, given a set $Msubsetmathbb{R}^n$, every point $mathbf{a}inmathbb{R}^n$ is either an interior, exterior or boundary point of $M$.
I recognize this must be true for $n = 1, 2, 3$ for geometric reasons but fail to see why it holds in $mathbb{R}^n$ where $n>3$. I tried proving it using the definitions of interior, exterior and boundary points in $mathbb{R}^n$ but got nowhere.
multivariable-calculus
asked Jan 22 at 13:12
S. DoeS. Doe
62
$endgroup$
$begingroup$
Just wondering, what could the point be other than the 3 options, interior, exterior or boundary?
$endgroup$
– denklo
Jan 22 at 13:15
2
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Strange as the definitions are independent from the dimension. Hence the proves are valid whatever the dimension is. What definitions are you using for interior, exterior and boundary?
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– mathcounterexamples.net
Jan 22 at 13:16
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@mathcounterexamples.net I'm guessing the standardized ones. $mathbf{a}$ is an interior point of $M$ if there is an open ball centered at $mathbf{a}$ that lies entirely within $M$; $mathbf{a}$ is an exterior point of $M$ if there is an open ball centered around $mathbf{a}$ that lies entirely within the complement $complement M$ of $M$; $mathbf{a}$ is a boundary point of $M$ if every open ball centered at $mathbf{a}$ contains points from both $M$ and $complement M$.
$endgroup$
– S. Doe
Jan 22 at 13:26
$begingroup$
Your definitions look good, and are independent from the dimension. You just need to remind what is an open ball in $mathbb R^n$.
$endgroup$
– mathcounterexamples.net
Jan 22 at 13:36
$begingroup$
@mathcounterexamples.net I'm guessing I'm missing something obvious. I'll have another go at it.
$endgroup$
– S. Doe
Jan 22 at 13:42
|
show 2 more comments
$begingroup$
I'm currently taking a class on multivariable calculus. The textbook begins by defining interior, exterior and boundary points in $mathbb{R}^n$ and states that, given a set $Msubsetmathbb{R}^n$, every point $mathbf{a}inmathbb{R}^n$ is either an interior, exterior or boundary point of $M$.
I recognize this must be true for $n = 1, 2, 3$ for geometric reasons but fail to see why it holds in $mathbb{R}^n$ where $n>3$. I tried proving it using the definitions of interior, exterior and boundary points in $mathbb{R}^n$ but got nowhere.
multivariable-calculus
asked Jan 22 at 13:12
S. DoeS. Doe
62
$endgroup$
I'm currently taking a class on multivariable calculus. The textbook begins by defining interior, exterior and boundary points in $mathbb{R}^n$ and states that, given a set $Msubsetmathbb{R}^n$, every point $mathbf{a}inmathbb{R}^n$ is either an interior, exterior or boundary point of $M$.
I recognize this must be true for $n = 1, 2, 3$ for geometric reasons but fail to see why it holds in $mathbb{R}^n$ where $n>3$. I tried proving it using the definitions of interior, exterior and boundary points in $mathbb{R}^n$ but got nowhere.
multivariable-calculus
multivariable-calculus
asked Jan 22 at 13:12
S. DoeS. Doe
62
asked Jan 22 at 13:12
S. DoeS. Doe
62
asked Jan 22 at 13:12
S. DoeS. Doe
62
asked Jan 22 at 13:12
S. DoeS. Doe
62
asked Jan 22 at 13:12
S. DoeS. Doe
62
62
$begingroup$
Just wondering, what could the point be other than the 3 options, interior, exterior or boundary?
$endgroup$
– denklo
Jan 22 at 13:15
2
$begingroup$
Strange as the definitions are independent from the dimension. Hence the proves are valid whatever the dimension is. What definitions are you using for interior, exterior and boundary?
$endgroup$
– mathcounterexamples.net
Jan 22 at 13:16
$begingroup$
@mathcounterexamples.net I'm guessing the standardized ones. $mathbf{a}$ is an interior point of $M$ if there is an open ball centered at $mathbf{a}$ that lies entirely within $M$; $mathbf{a}$ is an exterior point of $M$ if there is an open ball centered around $mathbf{a}$ that lies entirely within the complement $complement M$ of $M$; $mathbf{a}$ is a boundary point of $M$ if every open ball centered at $mathbf{a}$ contains points from both $M$ and $complement M$.
$endgroup$
– S. Doe
Jan 22 at 13:26
$begingroup$
Your definitions look good, and are independent from the dimension. You just need to remind what is an open ball in $mathbb R^n$.
$endgroup$
– mathcounterexamples.net
Jan 22 at 13:36
$begingroup$
@mathcounterexamples.net I'm guessing I'm missing something obvious. I'll have another go at it.
$endgroup$
– S. Doe
Jan 22 at 13:42
|
show 2 more comments
$begingroup$
Just wondering, what could the point be other than the 3 options, interior, exterior or boundary?
$endgroup$
– denklo
Jan 22 at 13:15
2
$begingroup$
Strange as the definitions are independent from the dimension. Hence the proves are valid whatever the dimension is. What definitions are you using for interior, exterior and boundary?
$endgroup$
– mathcounterexamples.net
Jan 22 at 13:16
$begingroup$
@mathcounterexamples.net I'm guessing the standardized ones. $mathbf{a}$ is an interior point of $M$ if there is an open ball centered at $mathbf{a}$ that lies entirely within $M$; $mathbf{a}$ is an exterior point of $M$ if there is an open ball centered around $mathbf{a}$ that lies entirely within the complement $complement M$ of $M$; $mathbf{a}$ is a boundary point of $M$ if every open ball centered at $mathbf{a}$ contains points from both $M$ and $complement M$.
$endgroup$
– S. Doe
Jan 22 at 13:26
$begingroup$
Your definitions look good, and are independent from the dimension. You just need to remind what is an open ball in $mathbb R^n$.
$endgroup$
– mathcounterexamples.net
Jan 22 at 13:36
$begingroup$
@mathcounterexamples.net I'm guessing I'm missing something obvious. I'll have another go at it.
$endgroup$
– S. Doe
Jan 22 at 13:42
$begingroup$
Just wondering, what could the point be other than the 3 options, interior, exterior or boundary?
$endgroup$
– denklo
Jan 22 at 13:15
$begingroup$
Just wondering, what could the point be other than the 3 options, interior, exterior or boundary?
$endgroup$
– denklo
Jan 22 at 13:15
2
2
$begingroup$
Strange as the definitions are independent from the dimension. Hence the proves are valid whatever the dimension is. What definitions are you using for interior, exterior and boundary?
$endgroup$
– mathcounterexamples.net
Jan 22 at 13:16
$begingroup$
Strange as the definitions are independent from the dimension. Hence the proves are valid whatever the dimension is. What definitions are you using for interior, exterior and boundary?
$endgroup$
– mathcounterexamples.net
Jan 22 at 13:16
$begingroup$
@mathcounterexamples.net I'm guessing the standardized ones. $mathbf{a}$ is an interior point of $M$ if there is an open ball centered at $mathbf{a}$ that lies entirely within $M$; $mathbf{a}$ is an exterior point of $M$ if there is an open ball centered around $mathbf{a}$ that lies entirely within the complement $complement M$ of $M$; $mathbf{a}$ is a boundary point of $M$ if every open ball centered at $mathbf{a}$ contains points from both $M$ and $complement M$.
$endgroup$
– S. Doe
Jan 22 at 13:26
$begingroup$
@mathcounterexamples.net I'm guessing the standardized ones. $mathbf{a}$ is an interior point of $M$ if there is an open ball centered at $mathbf{a}$ that lies entirely within $M$; $mathbf{a}$ is an exterior point of $M$ if there is an open ball centered around $mathbf{a}$ that lies entirely within the complement $complement M$ of $M$; $mathbf{a}$ is a boundary point of $M$ if every open ball centered at $mathbf{a}$ contains points from both $M$ and $complement M$.
$endgroup$
– S. Doe
Jan 22 at 13:26
$begingroup$
Your definitions look good, and are independent from the dimension. You just need to remind what is an open ball in $mathbb R^n$.
$endgroup$
– mathcounterexamples.net
Jan 22 at 13:36
$begingroup$
Your definitions look good, and are independent from the dimension. You just need to remind what is an open ball in $mathbb R^n$.
$endgroup$
– mathcounterexamples.net
Jan 22 at 13:36
$begingroup$
@mathcounterexamples.net I'm guessing I'm missing something obvious. I'll have another go at it.
$endgroup$
– S. Doe
Jan 22 at 13:42
$begingroup$
@mathcounterexamples.net I'm guessing I'm missing something obvious. I'll have another go at it.
$endgroup$
– S. Doe
Jan 22 at 13:42
|
show 2 more comments
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$begingroup$
Just wondering, what could the point be other than the 3 options, interior, exterior or boundary?
$endgroup$
– denklo
Jan 22 at 13:15
2
$begingroup$
Strange as the definitions are independent from the dimension. Hence the proves are valid whatever the dimension is. What definitions are you using for interior, exterior and boundary?
$endgroup$
– mathcounterexamples.net
Jan 22 at 13:16
$begingroup$
@mathcounterexamples.net I'm guessing the standardized ones. $mathbf{a}$ is an interior point of $M$ if there is an open ball centered at $mathbf{a}$ that lies entirely within $M$; $mathbf{a}$ is an exterior point of $M$ if there is an open ball centered around $mathbf{a}$ that lies entirely within the complement $complement M$ of $M$; $mathbf{a}$ is a boundary point of $M$ if every open ball centered at $mathbf{a}$ contains points from both $M$ and $complement M$.
$endgroup$
– S. Doe
Jan 22 at 13:26
$begingroup$
Your definitions look good, and are independent from the dimension. You just need to remind what is an open ball in $mathbb R^n$.
$endgroup$
– mathcounterexamples.net
Jan 22 at 13:36
$begingroup$
@mathcounterexamples.net I'm guessing I'm missing something obvious. I'll have another go at it.
$endgroup$
– S. Doe
Jan 22 at 13:42