Confused about open and closed set in $mathbb{R}^2$
$begingroup$
While this question is basically meant for all $mathbb{R}^n$ where $r > 1$, I suppose it would be the easiest to grasp the concept in $mathbb{R}^2$ and conclude the analog to $mathbb{R}^2$.
So I was wondering since we just started out with Topology, how to find out if a set is closed or open in $mathbb{R}^2$.
Let's say I have a set $A subset mathbb{R}$ defined as:
$$A=(0,1) times [0,1]$$
Since I now have a set "composed" by an open and a closed set I wonder if $A$ is now called open or closed.
Or is this a case where it neither is open nor closed?
general-topology
$endgroup$
add a comment |
$begingroup$
While this question is basically meant for all $mathbb{R}^n$ where $r > 1$, I suppose it would be the easiest to grasp the concept in $mathbb{R}^2$ and conclude the analog to $mathbb{R}^2$.
So I was wondering since we just started out with Topology, how to find out if a set is closed or open in $mathbb{R}^2$.
Let's say I have a set $A subset mathbb{R}$ defined as:
$$A=(0,1) times [0,1]$$
Since I now have a set "composed" by an open and a closed set I wonder if $A$ is now called open or closed.
Or is this a case where it neither is open nor closed?
general-topology
$endgroup$
1
$begingroup$
It is neither open nor closed. This is true of most subsets in a sense.
$endgroup$
– Matt Samuel
Dec 15 '18 at 12:53
add a comment |
$begingroup$
While this question is basically meant for all $mathbb{R}^n$ where $r > 1$, I suppose it would be the easiest to grasp the concept in $mathbb{R}^2$ and conclude the analog to $mathbb{R}^2$.
So I was wondering since we just started out with Topology, how to find out if a set is closed or open in $mathbb{R}^2$.
Let's say I have a set $A subset mathbb{R}$ defined as:
$$A=(0,1) times [0,1]$$
Since I now have a set "composed" by an open and a closed set I wonder if $A$ is now called open or closed.
Or is this a case where it neither is open nor closed?
general-topology
$endgroup$
While this question is basically meant for all $mathbb{R}^n$ where $r > 1$, I suppose it would be the easiest to grasp the concept in $mathbb{R}^2$ and conclude the analog to $mathbb{R}^2$.
So I was wondering since we just started out with Topology, how to find out if a set is closed or open in $mathbb{R}^2$.
Let's say I have a set $A subset mathbb{R}$ defined as:
$$A=(0,1) times [0,1]$$
Since I now have a set "composed" by an open and a closed set I wonder if $A$ is now called open or closed.
Or is this a case where it neither is open nor closed?
general-topology
general-topology
edited Dec 15 '18 at 13:08
Andrés E. Caicedo
65.4k8158249
65.4k8158249
asked Dec 15 '18 at 12:51
MLKMLK
80112
80112
1
$begingroup$
It is neither open nor closed. This is true of most subsets in a sense.
$endgroup$
– Matt Samuel
Dec 15 '18 at 12:53
add a comment |
1
$begingroup$
It is neither open nor closed. This is true of most subsets in a sense.
$endgroup$
– Matt Samuel
Dec 15 '18 at 12:53
1
1
$begingroup$
It is neither open nor closed. This is true of most subsets in a sense.
$endgroup$
– Matt Samuel
Dec 15 '18 at 12:53
$begingroup$
It is neither open nor closed. This is true of most subsets in a sense.
$endgroup$
– Matt Samuel
Dec 15 '18 at 12:53
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
- Since $(0,0)notin A$ but every open ball centered at $(0,0)$ contains points of $A$, $A$ is not closed.
- Since $left(frac12,0right)in A$ but every open ball centered at $left(frac12,0right)in A$ contains points out of $A$, $A$ is not open.
Therefore, $A$ is neither closed nor open.
$endgroup$
$begingroup$
Interesting. Thank you. So basically for every $mathbb{R}^n$ one just has to look at every set that is making up the set $A$ and if one is open while one is closed it is neither the one nor the other
$endgroup$
– MLK
Dec 15 '18 at 12:58
1
$begingroup$
Yes, but that's an almost useless piece of information, since very few subsets of $mathbb{R}^2$ can be expressed as the product of two subsets of $mathbb R$.
$endgroup$
– José Carlos Santos
Dec 15 '18 at 13:45
add a comment |
$begingroup$
This set $(0,1) times [0,1]$ is not closed and not open.
Not closed e.g. because $(frac{1}{n}, frac{1}{n})$ is a sequence in it conervging to $(0,0) notin A$. Not open e.g. because $(frac{1}{2}, -frac{1}{n})$ is a sequence converging to $(frac12,0)$ in $A$ but all of whose points lie outside $A$.
Another way: projections are open maps so if $A$ were open, so would $pi_2[A]= [0,1]$ be in $mathbb{R}$, which it is not. If $A$ were closed, it would be compact (closed and bounded in $mathbb{R}^2$) but $pi_1[A] = (0,1)$ is not compact (not even closed).
$endgroup$
add a comment |
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2 Answers
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2 Answers
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$begingroup$
- Since $(0,0)notin A$ but every open ball centered at $(0,0)$ contains points of $A$, $A$ is not closed.
- Since $left(frac12,0right)in A$ but every open ball centered at $left(frac12,0right)in A$ contains points out of $A$, $A$ is not open.
Therefore, $A$ is neither closed nor open.
$endgroup$
$begingroup$
Interesting. Thank you. So basically for every $mathbb{R}^n$ one just has to look at every set that is making up the set $A$ and if one is open while one is closed it is neither the one nor the other
$endgroup$
– MLK
Dec 15 '18 at 12:58
1
$begingroup$
Yes, but that's an almost useless piece of information, since very few subsets of $mathbb{R}^2$ can be expressed as the product of two subsets of $mathbb R$.
$endgroup$
– José Carlos Santos
Dec 15 '18 at 13:45
add a comment |
$begingroup$
- Since $(0,0)notin A$ but every open ball centered at $(0,0)$ contains points of $A$, $A$ is not closed.
- Since $left(frac12,0right)in A$ but every open ball centered at $left(frac12,0right)in A$ contains points out of $A$, $A$ is not open.
Therefore, $A$ is neither closed nor open.
$endgroup$
$begingroup$
Interesting. Thank you. So basically for every $mathbb{R}^n$ one just has to look at every set that is making up the set $A$ and if one is open while one is closed it is neither the one nor the other
$endgroup$
– MLK
Dec 15 '18 at 12:58
1
$begingroup$
Yes, but that's an almost useless piece of information, since very few subsets of $mathbb{R}^2$ can be expressed as the product of two subsets of $mathbb R$.
$endgroup$
– José Carlos Santos
Dec 15 '18 at 13:45
add a comment |
$begingroup$
- Since $(0,0)notin A$ but every open ball centered at $(0,0)$ contains points of $A$, $A$ is not closed.
- Since $left(frac12,0right)in A$ but every open ball centered at $left(frac12,0right)in A$ contains points out of $A$, $A$ is not open.
Therefore, $A$ is neither closed nor open.
$endgroup$
- Since $(0,0)notin A$ but every open ball centered at $(0,0)$ contains points of $A$, $A$ is not closed.
- Since $left(frac12,0right)in A$ but every open ball centered at $left(frac12,0right)in A$ contains points out of $A$, $A$ is not open.
Therefore, $A$ is neither closed nor open.
answered Dec 15 '18 at 12:55
José Carlos SantosJosé Carlos Santos
160k22127232
160k22127232
$begingroup$
Interesting. Thank you. So basically for every $mathbb{R}^n$ one just has to look at every set that is making up the set $A$ and if one is open while one is closed it is neither the one nor the other
$endgroup$
– MLK
Dec 15 '18 at 12:58
1
$begingroup$
Yes, but that's an almost useless piece of information, since very few subsets of $mathbb{R}^2$ can be expressed as the product of two subsets of $mathbb R$.
$endgroup$
– José Carlos Santos
Dec 15 '18 at 13:45
add a comment |
$begingroup$
Interesting. Thank you. So basically for every $mathbb{R}^n$ one just has to look at every set that is making up the set $A$ and if one is open while one is closed it is neither the one nor the other
$endgroup$
– MLK
Dec 15 '18 at 12:58
1
$begingroup$
Yes, but that's an almost useless piece of information, since very few subsets of $mathbb{R}^2$ can be expressed as the product of two subsets of $mathbb R$.
$endgroup$
– José Carlos Santos
Dec 15 '18 at 13:45
$begingroup$
Interesting. Thank you. So basically for every $mathbb{R}^n$ one just has to look at every set that is making up the set $A$ and if one is open while one is closed it is neither the one nor the other
$endgroup$
– MLK
Dec 15 '18 at 12:58
$begingroup$
Interesting. Thank you. So basically for every $mathbb{R}^n$ one just has to look at every set that is making up the set $A$ and if one is open while one is closed it is neither the one nor the other
$endgroup$
– MLK
Dec 15 '18 at 12:58
1
1
$begingroup$
Yes, but that's an almost useless piece of information, since very few subsets of $mathbb{R}^2$ can be expressed as the product of two subsets of $mathbb R$.
$endgroup$
– José Carlos Santos
Dec 15 '18 at 13:45
$begingroup$
Yes, but that's an almost useless piece of information, since very few subsets of $mathbb{R}^2$ can be expressed as the product of two subsets of $mathbb R$.
$endgroup$
– José Carlos Santos
Dec 15 '18 at 13:45
add a comment |
$begingroup$
This set $(0,1) times [0,1]$ is not closed and not open.
Not closed e.g. because $(frac{1}{n}, frac{1}{n})$ is a sequence in it conervging to $(0,0) notin A$. Not open e.g. because $(frac{1}{2}, -frac{1}{n})$ is a sequence converging to $(frac12,0)$ in $A$ but all of whose points lie outside $A$.
Another way: projections are open maps so if $A$ were open, so would $pi_2[A]= [0,1]$ be in $mathbb{R}$, which it is not. If $A$ were closed, it would be compact (closed and bounded in $mathbb{R}^2$) but $pi_1[A] = (0,1)$ is not compact (not even closed).
$endgroup$
add a comment |
$begingroup$
This set $(0,1) times [0,1]$ is not closed and not open.
Not closed e.g. because $(frac{1}{n}, frac{1}{n})$ is a sequence in it conervging to $(0,0) notin A$. Not open e.g. because $(frac{1}{2}, -frac{1}{n})$ is a sequence converging to $(frac12,0)$ in $A$ but all of whose points lie outside $A$.
Another way: projections are open maps so if $A$ were open, so would $pi_2[A]= [0,1]$ be in $mathbb{R}$, which it is not. If $A$ were closed, it would be compact (closed and bounded in $mathbb{R}^2$) but $pi_1[A] = (0,1)$ is not compact (not even closed).
$endgroup$
add a comment |
$begingroup$
This set $(0,1) times [0,1]$ is not closed and not open.
Not closed e.g. because $(frac{1}{n}, frac{1}{n})$ is a sequence in it conervging to $(0,0) notin A$. Not open e.g. because $(frac{1}{2}, -frac{1}{n})$ is a sequence converging to $(frac12,0)$ in $A$ but all of whose points lie outside $A$.
Another way: projections are open maps so if $A$ were open, so would $pi_2[A]= [0,1]$ be in $mathbb{R}$, which it is not. If $A$ were closed, it would be compact (closed and bounded in $mathbb{R}^2$) but $pi_1[A] = (0,1)$ is not compact (not even closed).
$endgroup$
This set $(0,1) times [0,1]$ is not closed and not open.
Not closed e.g. because $(frac{1}{n}, frac{1}{n})$ is a sequence in it conervging to $(0,0) notin A$. Not open e.g. because $(frac{1}{2}, -frac{1}{n})$ is a sequence converging to $(frac12,0)$ in $A$ but all of whose points lie outside $A$.
Another way: projections are open maps so if $A$ were open, so would $pi_2[A]= [0,1]$ be in $mathbb{R}$, which it is not. If $A$ were closed, it would be compact (closed and bounded in $mathbb{R}^2$) but $pi_1[A] = (0,1)$ is not compact (not even closed).
answered Dec 15 '18 at 12:58
Henno BrandsmaHenno Brandsma
109k347114
109k347114
add a comment |
add a comment |
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$begingroup$
It is neither open nor closed. This is true of most subsets in a sense.
$endgroup$
– Matt Samuel
Dec 15 '18 at 12:53