Mixing
Is $[1,5]$ an open set?
$begingroup$
Wikipedia page for open set gives two definitions for the open set:
First
A subset $U$ of the Euclidean $n$-space $mathbb R^n$ is called open if, given any point $x$ in $U$, there exists a real number $ε > 0$ such that, given any point $y$ in $mathbb R^n$ whose Euclidean distance from $x$ is smaller than $ε$, $y$ also belongs to $U$.
Second
All elements of $T$ where
The trivial subsets are in T.
Whenever sets $A$ and $B$ are in $T$, then so is $A$ intersection $B$.
Whenever two or more sets are in $T$, then so is their union
Consider the topology of $mathbb R$.
Now, according to the first definition, $[1,5]$ is not open because there exists no ε>0 such that $5+ε$ is in $[1,5]$.
However, we now let $T$ be the family of all subsets of $mathbb R$.
We can see that $T$ satisfies all $3$ conditions outlined in the second definition. Thus, all elements of $T$ are open, which includes $[1,5]$.
Now, obviously one of these must be wrong. It should be the second one, but I don't see why exactly. Any help is appreciated.
general-topology
edited Jan 27 at 15:24
Exp ikx
4489
asked Jan 27 at 15:20
Star Platinum ZA WARUDOStar Platinum ZA WARUDO
349112
$endgroup$
add a comment |
$begingroup$
Wikipedia page for open set gives two definitions for the open set:
First
A subset $U$ of the Euclidean $n$-space $mathbb R^n$ is called open if, given any point $x$ in $U$, there exists a real number $ε > 0$ such that, given any point $y$ in $mathbb R^n$ whose Euclidean distance from $x$ is smaller than $ε$, $y$ also belongs to $U$.
Second
All elements of $T$ where
The trivial subsets are in T.
Whenever sets $A$ and $B$ are in $T$, then so is $A$ intersection $B$.
Whenever two or more sets are in $T$, then so is their union
Consider the topology of $mathbb R$.
Now, according to the first definition, $[1,5]$ is not open because there exists no ε>0 such that $5+ε$ is in $[1,5]$.
However, we now let $T$ be the family of all subsets of $mathbb R$.
We can see that $T$ satisfies all $3$ conditions outlined in the second definition. Thus, all elements of $T$ are open, which includes $[1,5]$.
Now, obviously one of these must be wrong. It should be the second one, but I don't see why exactly. Any help is appreciated.
general-topology
edited Jan 27 at 15:24
Exp ikx
4489
asked Jan 27 at 15:20
Star Platinum ZA WARUDOStar Platinum ZA WARUDO
349112
$endgroup$
2
$begingroup$
The notion of open sets varies along with the underground topology.
$endgroup$
– xbh
Jan 27 at 15:21
$begingroup$
So, does that mean that both the results are correct in their respective contexts?
$endgroup$
– Star Platinum ZA WARUDO
Jan 27 at 15:23
$begingroup$
In this case, yes.
$endgroup$
– xbh
Jan 27 at 15:24
add a comment |
$begingroup$
Wikipedia page for open set gives two definitions for the open set:
First
A subset $U$ of the Euclidean $n$-space $mathbb R^n$ is called open if, given any point $x$ in $U$, there exists a real number $ε > 0$ such that, given any point $y$ in $mathbb R^n$ whose Euclidean distance from $x$ is smaller than $ε$, $y$ also belongs to $U$.
Second
All elements of $T$ where
The trivial subsets are in T.
Whenever sets $A$ and $B$ are in $T$, then so is $A$ intersection $B$.
Whenever two or more sets are in $T$, then so is their union
Consider the topology of $mathbb R$.
Now, according to the first definition, $[1,5]$ is not open because there exists no ε>0 such that $5+ε$ is in $[1,5]$.
However, we now let $T$ be the family of all subsets of $mathbb R$.
We can see that $T$ satisfies all $3$ conditions outlined in the second definition. Thus, all elements of $T$ are open, which includes $[1,5]$.
Now, obviously one of these must be wrong. It should be the second one, but I don't see why exactly. Any help is appreciated.
general-topology
edited Jan 27 at 15:24
Exp ikx
4489
asked Jan 27 at 15:20
Star Platinum ZA WARUDOStar Platinum ZA WARUDO
349112
$endgroup$
Wikipedia page for open set gives two definitions for the open set:
First
A subset $U$ of the Euclidean $n$-space $mathbb R^n$ is called open if, given any point $x$ in $U$, there exists a real number $ε > 0$ such that, given any point $y$ in $mathbb R^n$ whose Euclidean distance from $x$ is smaller than $ε$, $y$ also belongs to $U$.
Second
All elements of $T$ where
The trivial subsets are in T.
Whenever sets $A$ and $B$ are in $T$, then so is $A$ intersection $B$.
Whenever two or more sets are in $T$, then so is their union
Consider the topology of $mathbb R$.
Now, according to the first definition, $[1,5]$ is not open because there exists no ε>0 such that $5+ε$ is in $[1,5]$.
However, we now let $T$ be the family of all subsets of $mathbb R$.
We can see that $T$ satisfies all $3$ conditions outlined in the second definition. Thus, all elements of $T$ are open, which includes $[1,5]$.
Now, obviously one of these must be wrong. It should be the second one, but I don't see why exactly. Any help is appreciated.
general-topology
general-topology
edited Jan 27 at 15:24
Exp ikx
4489
asked Jan 27 at 15:20
Star Platinum ZA WARUDOStar Platinum ZA WARUDO
349112
edited Jan 27 at 15:24
Exp ikx
4489
asked Jan 27 at 15:20
Star Platinum ZA WARUDOStar Platinum ZA WARUDO
349112
edited Jan 27 at 15:24
Exp ikx
4489
edited Jan 27 at 15:24
Exp ikx
4489
edited Jan 27 at 15:24
Exp ikx
4489
4489
asked Jan 27 at 15:20
Star Platinum ZA WARUDOStar Platinum ZA WARUDO
349112
asked Jan 27 at 15:20
Star Platinum ZA WARUDOStar Platinum ZA WARUDO
349112
asked Jan 27 at 15:20
Star Platinum ZA WARUDOStar Platinum ZA WARUDO
349112
349112
2
$begingroup$
The notion of open sets varies along with the underground topology.
$endgroup$
– xbh
Jan 27 at 15:21
$begingroup$
So, does that mean that both the results are correct in their respective contexts?
$endgroup$
– Star Platinum ZA WARUDO
Jan 27 at 15:23
$begingroup$
In this case, yes.
$endgroup$
– xbh
Jan 27 at 15:24
add a comment |
2
$begingroup$
The notion of open sets varies along with the underground topology.
$endgroup$
– xbh
Jan 27 at 15:21
$begingroup$
So, does that mean that both the results are correct in their respective contexts?
$endgroup$
– Star Platinum ZA WARUDO
Jan 27 at 15:23
$begingroup$
In this case, yes.
$endgroup$
– xbh
Jan 27 at 15:24
2
2
$begingroup$
The notion of open sets varies along with the underground topology.
$endgroup$
– xbh
Jan 27 at 15:21
$begingroup$
The notion of open sets varies along with the underground topology.
$endgroup$
– xbh
Jan 27 at 15:21
$begingroup$
So, does that mean that both the results are correct in their respective contexts?
$endgroup$
– Star Platinum ZA WARUDO
Jan 27 at 15:23
$begingroup$
So, does that mean that both the results are correct in their respective contexts?
$endgroup$
– Star Platinum ZA WARUDO
Jan 27 at 15:23
$begingroup$
In this case, yes.
$endgroup$
– xbh
Jan 27 at 15:24
$begingroup$
In this case, yes.
$endgroup$
– xbh
Jan 27 at 15:24
add a comment |
1 Answer
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The first definition uses the standard topology on $mathbb{R}$, while the second is the more general definition of a topology. In your example, you simply created a different topology (the discrete topology) on $mathbb{R}$.
answered Jan 27 at 15:25
MetricMetric
1,22659
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add a comment |
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$begingroup$
The first definition uses the standard topology on $mathbb{R}$, while the second is the more general definition of a topology. In your example, you simply created a different topology (the discrete topology) on $mathbb{R}$.
answered Jan 27 at 15:25
MetricMetric
1,22659
$endgroup$
add a comment |
$begingroup$
The first definition uses the standard topology on $mathbb{R}$, while the second is the more general definition of a topology. In your example, you simply created a different topology (the discrete topology) on $mathbb{R}$.
answered Jan 27 at 15:25
MetricMetric
1,22659
$endgroup$
add a comment |
$begingroup$
The first definition uses the standard topology on $mathbb{R}$, while the second is the more general definition of a topology. In your example, you simply created a different topology (the discrete topology) on $mathbb{R}$.
answered Jan 27 at 15:25
MetricMetric
1,22659
$endgroup$
The first definition uses the standard topology on $mathbb{R}$, while the second is the more general definition of a topology. In your example, you simply created a different topology (the discrete topology) on $mathbb{R}$.
answered Jan 27 at 15:25
MetricMetric
1,22659
answered Jan 27 at 15:25
MetricMetric
1,22659
answered Jan 27 at 15:25
MetricMetric
1,22659
answered Jan 27 at 15:25
MetricMetric
1,22659
1,22659
add a comment |
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2
$begingroup$
The notion of open sets varies along with the underground topology.
$endgroup$
– xbh
Jan 27 at 15:21
$begingroup$
So, does that mean that both the results are correct in their respective contexts?
$endgroup$
– Star Platinum ZA WARUDO
Jan 27 at 15:23
$begingroup$
In this case, yes.
$endgroup$
– xbh
Jan 27 at 15:24