How to understand this example in Do Carmo?
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I'm reading the book $Riemannian$ $Geometry$ written by Do Carmo. Here is an example that I cannot understand the explanation he gave.
I really don't understand what he said about why $alpha$ is not an embedding...
No worry about my knowledge on topology, can anyone help me ''translate'' it to the common language that's easy to understand?
general-topology differential-geometry riemannian-geometry curves
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up vote
4
down vote
favorite
I'm reading the book $Riemannian$ $Geometry$ written by Do Carmo. Here is an example that I cannot understand the explanation he gave.
I really don't understand what he said about why $alpha$ is not an embedding...
No worry about my knowledge on topology, can anyone help me ''translate'' it to the common language that's easy to understand?
general-topology differential-geometry riemannian-geometry curves
add a comment |
up vote
4
down vote
favorite
up vote
4
down vote
favorite
I'm reading the book $Riemannian$ $Geometry$ written by Do Carmo. Here is an example that I cannot understand the explanation he gave.
I really don't understand what he said about why $alpha$ is not an embedding...
No worry about my knowledge on topology, can anyone help me ''translate'' it to the common language that's easy to understand?
general-topology differential-geometry riemannian-geometry curves
I'm reading the book $Riemannian$ $Geometry$ written by Do Carmo. Here is an example that I cannot understand the explanation he gave.
I really don't understand what he said about why $alpha$ is not an embedding...
No worry about my knowledge on topology, can anyone help me ''translate'' it to the common language that's easy to understand?
general-topology differential-geometry riemannian-geometry curves
general-topology differential-geometry riemannian-geometry curves
edited 12 hours ago
José Carlos Santos
137k17109199
137k17109199
asked 12 hours ago
user450201
597
597
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
8
down vote
accepted
The set $C=alphabigl((-3,0)bigr)$ has two topologies:
- the topology it inherits from the usual topology in $mathbbR^2$: a set $Asubset C$ is open if there is an open subset of $A^star$ of $mathbbR^2$ such that $A=A^starcap C$.
- the topology in gets from $(-3,0)$: a set $Asubset C$ is open if there is an open subcet $A^star$ of $(-3,0)$ such that $A=alpha(A^star)$.
Then, Do Carmo explains why these two topologies are distinct: the second one is locally connected, whereas the first one is no.
1
Can I just say that (-3,0) is locally connected while the image of alpha is not, so alpha is not a homeomorphism?
– user450201
12 hours ago
That would be correct. On the other hand, that is what Do Carmo is claiming.
– José Carlos Santos
12 hours ago
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
8
down vote
accepted
The set $C=alphabigl((-3,0)bigr)$ has two topologies:
- the topology it inherits from the usual topology in $mathbbR^2$: a set $Asubset C$ is open if there is an open subset of $A^star$ of $mathbbR^2$ such that $A=A^starcap C$.
- the topology in gets from $(-3,0)$: a set $Asubset C$ is open if there is an open subcet $A^star$ of $(-3,0)$ such that $A=alpha(A^star)$.
Then, Do Carmo explains why these two topologies are distinct: the second one is locally connected, whereas the first one is no.
1
Can I just say that (-3,0) is locally connected while the image of alpha is not, so alpha is not a homeomorphism?
– user450201
12 hours ago
That would be correct. On the other hand, that is what Do Carmo is claiming.
– José Carlos Santos
12 hours ago
add a comment |
up vote
8
down vote
accepted
The set $C=alphabigl((-3,0)bigr)$ has two topologies:
- the topology it inherits from the usual topology in $mathbbR^2$: a set $Asubset C$ is open if there is an open subset of $A^star$ of $mathbbR^2$ such that $A=A^starcap C$.
- the topology in gets from $(-3,0)$: a set $Asubset C$ is open if there is an open subcet $A^star$ of $(-3,0)$ such that $A=alpha(A^star)$.
Then, Do Carmo explains why these two topologies are distinct: the second one is locally connected, whereas the first one is no.
1
Can I just say that (-3,0) is locally connected while the image of alpha is not, so alpha is not a homeomorphism?
– user450201
12 hours ago
That would be correct. On the other hand, that is what Do Carmo is claiming.
– José Carlos Santos
12 hours ago
add a comment |
up vote
8
down vote
accepted
up vote
8
down vote
accepted
The set $C=alphabigl((-3,0)bigr)$ has two topologies:
- the topology it inherits from the usual topology in $mathbbR^2$: a set $Asubset C$ is open if there is an open subset of $A^star$ of $mathbbR^2$ such that $A=A^starcap C$.
- the topology in gets from $(-3,0)$: a set $Asubset C$ is open if there is an open subcet $A^star$ of $(-3,0)$ such that $A=alpha(A^star)$.
Then, Do Carmo explains why these two topologies are distinct: the second one is locally connected, whereas the first one is no.
The set $C=alphabigl((-3,0)bigr)$ has two topologies:
- the topology it inherits from the usual topology in $mathbbR^2$: a set $Asubset C$ is open if there is an open subset of $A^star$ of $mathbbR^2$ such that $A=A^starcap C$.
- the topology in gets from $(-3,0)$: a set $Asubset C$ is open if there is an open subcet $A^star$ of $(-3,0)$ such that $A=alpha(A^star)$.
Then, Do Carmo explains why these two topologies are distinct: the second one is locally connected, whereas the first one is no.
edited 9 hours ago
answered 12 hours ago
José Carlos Santos
137k17109199
137k17109199
1
Can I just say that (-3,0) is locally connected while the image of alpha is not, so alpha is not a homeomorphism?
– user450201
12 hours ago
That would be correct. On the other hand, that is what Do Carmo is claiming.
– José Carlos Santos
12 hours ago
add a comment |
1
Can I just say that (-3,0) is locally connected while the image of alpha is not, so alpha is not a homeomorphism?
– user450201
12 hours ago
That would be correct. On the other hand, that is what Do Carmo is claiming.
– José Carlos Santos
12 hours ago
1
1
Can I just say that (-3,0) is locally connected while the image of alpha is not, so alpha is not a homeomorphism?
– user450201
12 hours ago
Can I just say that (-3,0) is locally connected while the image of alpha is not, so alpha is not a homeomorphism?
– user450201
12 hours ago
That would be correct. On the other hand, that is what Do Carmo is claiming.
– José Carlos Santos
12 hours ago
That would be correct. On the other hand, that is what Do Carmo is claiming.
– José Carlos Santos
12 hours ago
add a comment |
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