Where is my reasoning wrong?
$begingroup$
We know that components of locally connected sets are open. Now consider the space
$$X=bigcuplimits_{q in mathbb{Q}}^{} {(x,y) in mathbb{R}^2; x^2+y^2=q^2 }-{(0,0) }$$
Now this seems locally connected, since every point belongs to a circle and you can just take smaller and smaller arcs around that point. However the components are the circles $x^2+y^2=q^2$ which aren't open.
Edit: $q in mathbb{Q}^{+}$ not $q in mathbb{Q}$
general-topology connectedness
asked Jan 8 at 13:14
InsertNameHereInsertNameHere
553
$endgroup$
add a comment |
$begingroup$
We know that components of locally connected sets are open. Now consider the space
$$X=bigcuplimits_{q in mathbb{Q}}^{} {(x,y) in mathbb{R}^2; x^2+y^2=q^2 }-{(0,0) }$$
Now this seems locally connected, since every point belongs to a circle and you can just take smaller and smaller arcs around that point. However the components are the circles $x^2+y^2=q^2$ which aren't open.
Edit: $q in mathbb{Q}^{+}$ not $q in mathbb{Q}$
general-topology connectedness
asked Jan 8 at 13:14
InsertNameHereInsertNameHere
553
$endgroup$
1
$begingroup$
Why write an edit as a remark. Just edit. Also, you have 150 characters for the title space, how about a more descriptive title perhaps?
$endgroup$
– Asaf Karagila♦
Jan 8 at 13:37
add a comment |
$begingroup$
We know that components of locally connected sets are open. Now consider the space
$$X=bigcuplimits_{q in mathbb{Q}}^{} {(x,y) in mathbb{R}^2; x^2+y^2=q^2 }-{(0,0) }$$
Now this seems locally connected, since every point belongs to a circle and you can just take smaller and smaller arcs around that point. However the components are the circles $x^2+y^2=q^2$ which aren't open.
Edit: $q in mathbb{Q}^{+}$ not $q in mathbb{Q}$
general-topology connectedness
asked Jan 8 at 13:14
InsertNameHereInsertNameHere
553
$endgroup$
We know that components of locally connected sets are open. Now consider the space
$$X=bigcuplimits_{q in mathbb{Q}}^{} {(x,y) in mathbb{R}^2; x^2+y^2=q^2 }-{(0,0) }$$
Now this seems locally connected, since every point belongs to a circle and you can just take smaller and smaller arcs around that point. However the components are the circles $x^2+y^2=q^2$ which aren't open.
Edit: $q in mathbb{Q}^{+}$ not $q in mathbb{Q}$
general-topology connectedness
general-topology connectedness
asked Jan 8 at 13:14
InsertNameHereInsertNameHere
553
asked Jan 8 at 13:14
InsertNameHereInsertNameHere
553
edited Jan 8 at 13:27
InsertNameHere
asked Jan 8 at 13:14
InsertNameHereInsertNameHere
553
asked Jan 8 at 13:14
InsertNameHereInsertNameHere
553
asked Jan 8 at 13:14
InsertNameHereInsertNameHere
553
553
1
$begingroup$
Why write an edit as a remark. Just edit. Also, you have 150 characters for the title space, how about a more descriptive title perhaps?
$endgroup$
– Asaf Karagila♦
Jan 8 at 13:37
add a comment |
1
$begingroup$
Why write an edit as a remark. Just edit. Also, you have 150 characters for the title space, how about a more descriptive title perhaps?
$endgroup$
– Asaf Karagila♦
Jan 8 at 13:37
1
1
$begingroup$
Why write an edit as a remark. Just edit. Also, you have 150 characters for the title space, how about a more descriptive title perhaps?
$endgroup$
– Asaf Karagila♦
Jan 8 at 13:37
$begingroup$
Why write an edit as a remark. Just edit. Also, you have 150 characters for the title space, how about a more descriptive title perhaps?
$endgroup$
– Asaf Karagila♦
Jan 8 at 13:37
add a comment |
1 Answer
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$begingroup$
It’s not locally connected at all: any point on the circle has basic neighbourhoods that are intersection of open balls in the plane with $X$, and these will contain many points from other rational-radius circles. So your arcs aren’t neighbourhoods at all.
You treat the circles like they’re a topological sum, not as a subspace of the plane.
answered Jan 8 at 13:26
Henno BrandsmaHenno Brandsma
110k347117
$endgroup$
$begingroup$
Oh yeah, forgot the neighborhood basis elements need to be open. Thanks!
$endgroup$
– InsertNameHere
Jan 8 at 13:33
$begingroup$
@InsertNameHere they need to contain an open set, yes, but neighbourhoods need not be open.
$endgroup$
– Henno Brandsma
Jan 8 at 13:35
add a comment |
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1 Answer
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$begingroup$
It’s not locally connected at all: any point on the circle has basic neighbourhoods that are intersection of open balls in the plane with $X$, and these will contain many points from other rational-radius circles. So your arcs aren’t neighbourhoods at all.
You treat the circles like they’re a topological sum, not as a subspace of the plane.
answered Jan 8 at 13:26
Henno BrandsmaHenno Brandsma
110k347117
$endgroup$
$begingroup$
Oh yeah, forgot the neighborhood basis elements need to be open. Thanks!
$endgroup$
– InsertNameHere
Jan 8 at 13:33
$begingroup$
@InsertNameHere they need to contain an open set, yes, but neighbourhoods need not be open.
$endgroup$
– Henno Brandsma
Jan 8 at 13:35
add a comment |
$begingroup$
It’s not locally connected at all: any point on the circle has basic neighbourhoods that are intersection of open balls in the plane with $X$, and these will contain many points from other rational-radius circles. So your arcs aren’t neighbourhoods at all.
You treat the circles like they’re a topological sum, not as a subspace of the plane.
answered Jan 8 at 13:26
Henno BrandsmaHenno Brandsma
110k347117
$endgroup$
$begingroup$
Oh yeah, forgot the neighborhood basis elements need to be open. Thanks!
$endgroup$
– InsertNameHere
Jan 8 at 13:33
$begingroup$
@InsertNameHere they need to contain an open set, yes, but neighbourhoods need not be open.
$endgroup$
– Henno Brandsma
Jan 8 at 13:35
add a comment |
$begingroup$
It’s not locally connected at all: any point on the circle has basic neighbourhoods that are intersection of open balls in the plane with $X$, and these will contain many points from other rational-radius circles. So your arcs aren’t neighbourhoods at all.
You treat the circles like they’re a topological sum, not as a subspace of the plane.
answered Jan 8 at 13:26
Henno BrandsmaHenno Brandsma
110k347117
$endgroup$
It’s not locally connected at all: any point on the circle has basic neighbourhoods that are intersection of open balls in the plane with $X$, and these will contain many points from other rational-radius circles. So your arcs aren’t neighbourhoods at all.
You treat the circles like they’re a topological sum, not as a subspace of the plane.
answered Jan 8 at 13:26
Henno BrandsmaHenno Brandsma
110k347117
answered Jan 8 at 13:26
Henno BrandsmaHenno Brandsma
110k347117
answered Jan 8 at 13:26
Henno BrandsmaHenno Brandsma
110k347117
answered Jan 8 at 13:26
Henno BrandsmaHenno Brandsma
110k347117
110k347117
$begingroup$
Oh yeah, forgot the neighborhood basis elements need to be open. Thanks!
$endgroup$
– InsertNameHere
Jan 8 at 13:33
$begingroup$
@InsertNameHere they need to contain an open set, yes, but neighbourhoods need not be open.
$endgroup$
– Henno Brandsma
Jan 8 at 13:35
add a comment |
$begingroup$
Oh yeah, forgot the neighborhood basis elements need to be open. Thanks!
$endgroup$
– InsertNameHere
Jan 8 at 13:33
$begingroup$
@InsertNameHere they need to contain an open set, yes, but neighbourhoods need not be open.
$endgroup$
– Henno Brandsma
Jan 8 at 13:35
$begingroup$
Oh yeah, forgot the neighborhood basis elements need to be open. Thanks!
$endgroup$
– InsertNameHere
Jan 8 at 13:33
$begingroup$
Oh yeah, forgot the neighborhood basis elements need to be open. Thanks!
$endgroup$
– InsertNameHere
Jan 8 at 13:33
$begingroup$
@InsertNameHere they need to contain an open set, yes, but neighbourhoods need not be open.
$endgroup$
– Henno Brandsma
Jan 8 at 13:35
$begingroup$
@InsertNameHere they need to contain an open set, yes, but neighbourhoods need not be open.
$endgroup$
– Henno Brandsma
Jan 8 at 13:35
add a comment |
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$begingroup$
Why write an edit as a remark. Just edit. Also, you have 150 characters for the title space, how about a more descriptive title perhaps?
$endgroup$
– Asaf Karagila♦
Jan 8 at 13:37