Mixing
How is possible that those shapes are equivalent in topology?
Clash Royale CLAN TAG#URR8PPP
up vote
24
down vote
favorite
I recently started to study topology, I have no idea about the subject so my question could be very simple but I need a clear explanation. It is about the page number 19 of Introducton to Topology by Colin Adams and Robert Franzosa; it said that the shapes:
are equivalent in topology, but one has just one hole and the other has two. is possible to add holes or stick holes?
general-topology
edited Aug 25 at 19:05
Micah
28.5k1361101
asked Aug 25 at 19:00
José MarÃn
12916
add a comment |Â
up vote
24
down vote
favorite
I recently started to study topology, I have no idea about the subject so my question could be very simple but I need a clear explanation. It is about the page number 19 of Introducton to Topology by Colin Adams and Robert Franzosa; it said that the shapes:
are equivalent in topology, but one has just one hole and the other has two. is possible to add holes or stick holes?
general-topology
edited Aug 25 at 19:05
Micah
28.5k1361101
asked Aug 25 at 19:00
José MarÃn
12916
1
Each of the two holes in the sphere has two circular edges. But the first picture also has two circular edges.
– Michael Hardy
Aug 26 at 1:27
4
It is important to realize that these examples are NOT two dimensional surfaces, they are three dimensional solids. Imagine the first solid to be a deflated rubber bag which is then "blown up" to the round second solid.
– user247327
Aug 26 at 3:31
1
The drawing is perhaps not the best but hopefully the others explained it.
– Tom
Aug 27 at 9:50
add a comment |Â
up vote
24
down vote
favorite
up vote
24
down vote
favorite
I recently started to study topology, I have no idea about the subject so my question could be very simple but I need a clear explanation. It is about the page number 19 of Introducton to Topology by Colin Adams and Robert Franzosa; it said that the shapes:
are equivalent in topology, but one has just one hole and the other has two. is possible to add holes or stick holes?
general-topology
edited Aug 25 at 19:05
Micah
28.5k1361101
asked Aug 25 at 19:00
José MarÃn
12916
I recently started to study topology, I have no idea about the subject so my question could be very simple but I need a clear explanation. It is about the page number 19 of Introducton to Topology by Colin Adams and Robert Franzosa; it said that the shapes:
are equivalent in topology, but one has just one hole and the other has two. is possible to add holes or stick holes?
general-topology
edited Aug 25 at 19:05
Micah
28.5k1361101
asked Aug 25 at 19:00
José MarÃn
12916
edited Aug 25 at 19:05
Micah
28.5k1361101
edited Aug 25 at 19:05
Micah
28.5k1361101
edited Aug 25 at 19:05
Micah
28.5k1361101
28.5k1361101
asked Aug 25 at 19:00
José MarÃn
12916
asked Aug 25 at 19:00
José MarÃn
12916
asked Aug 25 at 19:00
José MarÃn
12916
12916
1
Each of the two holes in the sphere has two circular edges. But the first picture also has two circular edges.
– Michael Hardy
Aug 26 at 1:27
4
It is important to realize that these examples are NOT two dimensional surfaces, they are three dimensional solids. Imagine the first solid to be a deflated rubber bag which is then "blown up" to the round second solid.
– user247327
Aug 26 at 3:31
1
The drawing is perhaps not the best but hopefully the others explained it.
– Tom
Aug 27 at 9:50
add a comment |Â
1
Each of the two holes in the sphere has two circular edges. But the first picture also has two circular edges.
– Michael Hardy
Aug 26 at 1:27
4
It is important to realize that these examples are NOT two dimensional surfaces, they are three dimensional solids. Imagine the first solid to be a deflated rubber bag which is then "blown up" to the round second solid.
– user247327
Aug 26 at 3:31
1
The drawing is perhaps not the best but hopefully the others explained it.
– Tom
Aug 27 at 9:50
1
1
Each of the two holes in the sphere has two circular edges. But the first picture also has two circular edges.
– Michael Hardy
Aug 26 at 1:27
Each of the two holes in the sphere has two circular edges. But the first picture also has two circular edges.
– Michael Hardy
Aug 26 at 1:27
4
4
It is important to realize that these examples are NOT two dimensional surfaces, they are three dimensional solids. Imagine the first solid to be a deflated rubber bag which is then "blown up" to the round second solid.
– user247327
Aug 26 at 3:31
It is important to realize that these examples are NOT two dimensional surfaces, they are three dimensional solids. Imagine the first solid to be a deflated rubber bag which is then "blown up" to the round second solid.
– user247327
Aug 26 at 3:31
1
1
The drawing is perhaps not the best but hopefully the others explained it.
– Tom
Aug 27 at 9:50
The drawing is perhaps not the best but hopefully the others explained it.
– Tom
Aug 27 at 9:50
add a comment |Â
4 Answers
4
active
oldest
votes
up vote
54
down vote
accepted
Look a bit more closely at the second picture. There's a couple of little dotted lines connecting the two holes that may be a bit hard to see.
That is meant to convey the impression they are the two ends of a single, long, curved hole through the interior.
answered Aug 26 at 0:34
The_Sympathizer
6,5522241
28
I see the trench, but which one is the superlaser and which one is the exhaust port?
– Lamar Latrell
Aug 26 at 4:39
add a comment |Â
up vote
40
down vote
The "two holes" in that sphere are two ends of the same hole. (That is, if you drilled one hole all the way through a sphere, you would end up with something that looked very much like your picture.)
answered Aug 25 at 19:04
Micah
28.5k1361101
3
Or you inflated the flattened donut, which happens to have a weak, more redundant part.
– MathAsFun
Aug 25 at 19:07
4
It’s probably worth noting that in topology a “hole†is not a hole unless it creates an opening that passes entirely through the shape. What we might call a hole... in a wall for instance, after drilling a hole for a screw or something... is not actually a hole in topology.
– Fogmeister
Aug 26 at 6:47
add a comment |Â
up vote
20
down vote
You may also notice the tunel, which I agree with you it is not clear in this photo.
answered Aug 25 at 19:10
dmtri
774317
7
I think this answer captures the real problem, namely that, in the second picture, the tube inside the sphere, which connects the two holes, is represented by a pair of dashed lines that are so faint as to be almost invisible.
– Andreas Blass
Aug 25 at 20:28
@AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too.
– dmtri
Aug 26 at 6:10
add a comment |Â
up vote
6
down vote
The cuboid and the sphere are topological euvivalent. Drill a hole through each body as indicated by the arrow. The resulting bodies are still topological equivalent.
answered Aug 25 at 20:06
miracle173
7,17922247
add a comment |Â
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
54
down vote
accepted
Look a bit more closely at the second picture. There's a couple of little dotted lines connecting the two holes that may be a bit hard to see.
That is meant to convey the impression they are the two ends of a single, long, curved hole through the interior.
answered Aug 26 at 0:34
The_Sympathizer
6,5522241
28
I see the trench, but which one is the superlaser and which one is the exhaust port?
– Lamar Latrell
Aug 26 at 4:39
add a comment |Â
up vote
54
down vote
accepted
Look a bit more closely at the second picture. There's a couple of little dotted lines connecting the two holes that may be a bit hard to see.
That is meant to convey the impression they are the two ends of a single, long, curved hole through the interior.
answered Aug 26 at 0:34
The_Sympathizer
6,5522241
28
I see the trench, but which one is the superlaser and which one is the exhaust port?
– Lamar Latrell
Aug 26 at 4:39
add a comment |Â
up vote
54
down vote
accepted
up vote
54
down vote
accepted
Look a bit more closely at the second picture. There's a couple of little dotted lines connecting the two holes that may be a bit hard to see.
That is meant to convey the impression they are the two ends of a single, long, curved hole through the interior.
answered Aug 26 at 0:34
The_Sympathizer
6,5522241
Look a bit more closely at the second picture. There's a couple of little dotted lines connecting the two holes that may be a bit hard to see.
That is meant to convey the impression they are the two ends of a single, long, curved hole through the interior.
answered Aug 26 at 0:34
The_Sympathizer
6,5522241
answered Aug 26 at 0:34
The_Sympathizer
6,5522241
answered Aug 26 at 0:34
The_Sympathizer
6,5522241
answered Aug 26 at 0:34
The_Sympathizer
6,5522241
6,5522241
28
I see the trench, but which one is the superlaser and which one is the exhaust port?
– Lamar Latrell
Aug 26 at 4:39
add a comment |Â
28
I see the trench, but which one is the superlaser and which one is the exhaust port?
– Lamar Latrell
Aug 26 at 4:39
28
28
I see the trench, but which one is the superlaser and which one is the exhaust port?
– Lamar Latrell
Aug 26 at 4:39
I see the trench, but which one is the superlaser and which one is the exhaust port?
– Lamar Latrell
Aug 26 at 4:39
add a comment |Â
up vote
40
down vote
The "two holes" in that sphere are two ends of the same hole. (That is, if you drilled one hole all the way through a sphere, you would end up with something that looked very much like your picture.)
answered Aug 25 at 19:04
Micah
28.5k1361101
3
Or you inflated the flattened donut, which happens to have a weak, more redundant part.
– MathAsFun
Aug 25 at 19:07
4
It’s probably worth noting that in topology a “hole†is not a hole unless it creates an opening that passes entirely through the shape. What we might call a hole... in a wall for instance, after drilling a hole for a screw or something... is not actually a hole in topology.
– Fogmeister
Aug 26 at 6:47
add a comment |Â
up vote
40
down vote
The "two holes" in that sphere are two ends of the same hole. (That is, if you drilled one hole all the way through a sphere, you would end up with something that looked very much like your picture.)
answered Aug 25 at 19:04
Micah
28.5k1361101
3
Or you inflated the flattened donut, which happens to have a weak, more redundant part.
– MathAsFun
Aug 25 at 19:07
4
It’s probably worth noting that in topology a “hole†is not a hole unless it creates an opening that passes entirely through the shape. What we might call a hole... in a wall for instance, after drilling a hole for a screw or something... is not actually a hole in topology.
– Fogmeister
Aug 26 at 6:47
add a comment |Â
up vote
40
down vote
up vote
40
down vote
The "two holes" in that sphere are two ends of the same hole. (That is, if you drilled one hole all the way through a sphere, you would end up with something that looked very much like your picture.)
answered Aug 25 at 19:04
Micah
28.5k1361101
The "two holes" in that sphere are two ends of the same hole. (That is, if you drilled one hole all the way through a sphere, you would end up with something that looked very much like your picture.)
answered Aug 25 at 19:04
Micah
28.5k1361101
answered Aug 25 at 19:04
Micah
28.5k1361101
answered Aug 25 at 19:04
Micah
28.5k1361101
answered Aug 25 at 19:04
Micah
28.5k1361101
28.5k1361101
3
Or you inflated the flattened donut, which happens to have a weak, more redundant part.
– MathAsFun
Aug 25 at 19:07
4
It’s probably worth noting that in topology a “hole†is not a hole unless it creates an opening that passes entirely through the shape. What we might call a hole... in a wall for instance, after drilling a hole for a screw or something... is not actually a hole in topology.
– Fogmeister
Aug 26 at 6:47
add a comment |Â
3
Or you inflated the flattened donut, which happens to have a weak, more redundant part.
– MathAsFun
Aug 25 at 19:07
4
It’s probably worth noting that in topology a “hole†is not a hole unless it creates an opening that passes entirely through the shape. What we might call a hole... in a wall for instance, after drilling a hole for a screw or something... is not actually a hole in topology.
– Fogmeister
Aug 26 at 6:47
3
3
Or you inflated the flattened donut, which happens to have a weak, more redundant part.
– MathAsFun
Aug 25 at 19:07
Or you inflated the flattened donut, which happens to have a weak, more redundant part.
– MathAsFun
Aug 25 at 19:07
4
4
It’s probably worth noting that in topology a “hole†is not a hole unless it creates an opening that passes entirely through the shape. What we might call a hole... in a wall for instance, after drilling a hole for a screw or something... is not actually a hole in topology.
– Fogmeister
Aug 26 at 6:47
It’s probably worth noting that in topology a “hole†is not a hole unless it creates an opening that passes entirely through the shape. What we might call a hole... in a wall for instance, after drilling a hole for a screw or something... is not actually a hole in topology.
– Fogmeister
Aug 26 at 6:47
add a comment |Â
up vote
20
down vote
You may also notice the tunel, which I agree with you it is not clear in this photo.
answered Aug 25 at 19:10
dmtri
774317
7
I think this answer captures the real problem, namely that, in the second picture, the tube inside the sphere, which connects the two holes, is represented by a pair of dashed lines that are so faint as to be almost invisible.
– Andreas Blass
Aug 25 at 20:28
@AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too.
– dmtri
Aug 26 at 6:10
add a comment |Â
up vote
20
down vote
You may also notice the tunel, which I agree with you it is not clear in this photo.
answered Aug 25 at 19:10
dmtri
774317
7
I think this answer captures the real problem, namely that, in the second picture, the tube inside the sphere, which connects the two holes, is represented by a pair of dashed lines that are so faint as to be almost invisible.
– Andreas Blass
Aug 25 at 20:28
@AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too.
– dmtri
Aug 26 at 6:10
add a comment |Â
up vote
20
down vote
up vote
20
down vote
You may also notice the tunel, which I agree with you it is not clear in this photo.
answered Aug 25 at 19:10
dmtri
774317
You may also notice the tunel, which I agree with you it is not clear in this photo.
answered Aug 25 at 19:10
dmtri
774317
answered Aug 25 at 19:10
dmtri
774317
answered Aug 25 at 19:10
dmtri
774317
answered Aug 25 at 19:10
dmtri
774317
774317
7
I think this answer captures the real problem, namely that, in the second picture, the tube inside the sphere, which connects the two holes, is represented by a pair of dashed lines that are so faint as to be almost invisible.
– Andreas Blass
Aug 25 at 20:28
@AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too.
– dmtri
Aug 26 at 6:10
add a comment |Â
7
I think this answer captures the real problem, namely that, in the second picture, the tube inside the sphere, which connects the two holes, is represented by a pair of dashed lines that are so faint as to be almost invisible.
– Andreas Blass
Aug 25 at 20:28
@AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too.
– dmtri
Aug 26 at 6:10
7
7
I think this answer captures the real problem, namely that, in the second picture, the tube inside the sphere, which connects the two holes, is represented by a pair of dashed lines that are so faint as to be almost invisible.
– Andreas Blass
Aug 25 at 20:28
I think this answer captures the real problem, namely that, in the second picture, the tube inside the sphere, which connects the two holes, is represented by a pair of dashed lines that are so faint as to be almost invisible.
– Andreas Blass
Aug 25 at 20:28
@AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too.
– dmtri
Aug 26 at 6:10
@AndreasBlass, thanks for the comment, that way I managed to receive the "nice answer" badge. +1 from me too.
– dmtri
Aug 26 at 6:10
add a comment |Â
up vote
6
down vote
The cuboid and the sphere are topological euvivalent. Drill a hole through each body as indicated by the arrow. The resulting bodies are still topological equivalent.
answered Aug 25 at 20:06
miracle173
7,17922247
add a comment |Â
up vote
6
down vote
The cuboid and the sphere are topological euvivalent. Drill a hole through each body as indicated by the arrow. The resulting bodies are still topological equivalent.
answered Aug 25 at 20:06
miracle173
7,17922247
add a comment |Â
up vote
6
down vote
up vote
6
down vote
The cuboid and the sphere are topological euvivalent. Drill a hole through each body as indicated by the arrow. The resulting bodies are still topological equivalent.
answered Aug 25 at 20:06
miracle173
7,17922247
The cuboid and the sphere are topological euvivalent. Drill a hole through each body as indicated by the arrow. The resulting bodies are still topological equivalent.
answered Aug 25 at 20:06
miracle173
7,17922247
answered Aug 25 at 20:06
miracle173
7,17922247
answered Aug 25 at 20:06
miracle173
7,17922247
answered Aug 25 at 20:06
miracle173
7,17922247
7,17922247
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2894411%2fhow-is-possible-that-those-shapes-are-equivalent-in-topology%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
1
Each of the two holes in the sphere has two circular edges. But the first picture also has two circular edges.
– Michael Hardy
Aug 26 at 1:27
4
It is important to realize that these examples are NOT two dimensional surfaces, they are three dimensional solids. Imagine the first solid to be a deflated rubber bag which is then "blown up" to the round second solid.
– user247327
Aug 26 at 3:31
1
The drawing is perhaps not the best but hopefully the others explained it.
– Tom
Aug 27 at 9:50