Mixing
Intersection of subcomplexes
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Ok, so intuitively it's clear that the intersection of two subcomplexes of a CW-complex should be a subcomplex as well, but reading the inductive definition of a CW-complex, nowhere does it say that a cell should be attached to a whole other cell, that is to say: it seems to imply that i could, for example, attach a 2-cell to a point in the middle of a 1-cell as if there were a 0-cell there. But then the intersection of the 1-cell and the 2-cell in question would be a point that isn't a 0-cell, and therefore not a subcomplex.
Am I missing something from the definition?
cw-complexes
asked Aug 6 at 23:19
Santiago Bosch
406
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up vote
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down vote
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Ok, so intuitively it's clear that the intersection of two subcomplexes of a CW-complex should be a subcomplex as well, but reading the inductive definition of a CW-complex, nowhere does it say that a cell should be attached to a whole other cell, that is to say: it seems to imply that i could, for example, attach a 2-cell to a point in the middle of a 1-cell as if there were a 0-cell there. But then the intersection of the 1-cell and the 2-cell in question would be a point that isn't a 0-cell, and therefore not a subcomplex.
Am I missing something from the definition?
cw-complexes
asked Aug 6 at 23:19
Santiago Bosch
406
What is your definition of a CW complex?
– Sheel Stueber
Aug 7 at 1:25
the one that appears in Hatcher's "Algebraic Topology", you start by a discrete 0-skeleton and then attach cells via a map from the border of an n-disk to the n-1-skeleton. The thing is, that doesn't make it clear if the edge can go to any part of the skeleton or if it must attach to a full cell.
– Santiago Bosch
Aug 7 at 14:10
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Ok, so intuitively it's clear that the intersection of two subcomplexes of a CW-complex should be a subcomplex as well, but reading the inductive definition of a CW-complex, nowhere does it say that a cell should be attached to a whole other cell, that is to say: it seems to imply that i could, for example, attach a 2-cell to a point in the middle of a 1-cell as if there were a 0-cell there. But then the intersection of the 1-cell and the 2-cell in question would be a point that isn't a 0-cell, and therefore not a subcomplex.
Am I missing something from the definition?
cw-complexes
asked Aug 6 at 23:19
Santiago Bosch
406
Ok, so intuitively it's clear that the intersection of two subcomplexes of a CW-complex should be a subcomplex as well, but reading the inductive definition of a CW-complex, nowhere does it say that a cell should be attached to a whole other cell, that is to say: it seems to imply that i could, for example, attach a 2-cell to a point in the middle of a 1-cell as if there were a 0-cell there. But then the intersection of the 1-cell and the 2-cell in question would be a point that isn't a 0-cell, and therefore not a subcomplex.
Am I missing something from the definition?
cw-complexes
asked Aug 6 at 23:19
Santiago Bosch
406
asked Aug 6 at 23:19
Santiago Bosch
406
asked Aug 6 at 23:19
Santiago Bosch
406
asked Aug 6 at 23:19
Santiago Bosch
406
406
What is your definition of a CW complex?
– Sheel Stueber
Aug 7 at 1:25
the one that appears in Hatcher's "Algebraic Topology", you start by a discrete 0-skeleton and then attach cells via a map from the border of an n-disk to the n-1-skeleton. The thing is, that doesn't make it clear if the edge can go to any part of the skeleton or if it must attach to a full cell.
– Santiago Bosch
Aug 7 at 14:10
add a comment |Â
What is your definition of a CW complex?
– Sheel Stueber
Aug 7 at 1:25
the one that appears in Hatcher's "Algebraic Topology", you start by a discrete 0-skeleton and then attach cells via a map from the border of an n-disk to the n-1-skeleton. The thing is, that doesn't make it clear if the edge can go to any part of the skeleton or if it must attach to a full cell.
– Santiago Bosch
Aug 7 at 14:10
What is your definition of a CW complex?
– Sheel Stueber
Aug 7 at 1:25
What is your definition of a CW complex?
– Sheel Stueber
Aug 7 at 1:25
the one that appears in Hatcher's "Algebraic Topology", you start by a discrete 0-skeleton and then attach cells via a map from the border of an n-disk to the n-1-skeleton. The thing is, that doesn't make it clear if the edge can go to any part of the skeleton or if it must attach to a full cell.
– Santiago Bosch
Aug 7 at 14:10
the one that appears in Hatcher's "Algebraic Topology", you start by a discrete 0-skeleton and then attach cells via a map from the border of an n-disk to the n-1-skeleton. The thing is, that doesn't make it clear if the edge can go to any part of the skeleton or if it must attach to a full cell.
– Santiago Bosch
Aug 7 at 14:10
add a comment |Â
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What is your definition of a CW complex?
– Sheel Stueber
Aug 7 at 1:25
the one that appears in Hatcher's "Algebraic Topology", you start by a discrete 0-skeleton and then attach cells via a map from the border of an n-disk to the n-1-skeleton. The thing is, that doesn't make it clear if the edge can go to any part of the skeleton or if it must attach to a full cell.
– Santiago Bosch
Aug 7 at 14:10