How should I deal with point-set topology problems in algebraic topology?
$begingroup$
In algebraic topology, many point-set topology problems arise. For example, the product of quotient maps need not be a quotient map; the smash product may not be associative; the canonical map $Z^{Xtimes Y}cong(Z^Y)^X$ is not necessarily a homeomorphism...
It seems that many authors chooses to stay in the category of compactly generated (weak Hausdorff) spaces to remedy this. For example, in Chapter 5 of Algebraic Topology by J. P. May, he introduces this concept and states without proof some basic properties, and then he assumes all topological spaces are compactly generated in the remainder of the book.
On the other hand, some authors do not assume this. Then some restrictions are necessary. In this context local compactness frequently crops up.
Here are my questions:
- Which approach should I take, as a beginner in this subject who does not want to be overwhelmed by technicalities?
- I currently know nothing about compactly generated spaces. I really want to read the text by May, but he uses this throughout, making many of his assertions simply false for me without the necessary restrictions (for instance, a cofibration need not be closed). How should I deal with this problem?
- Are there any readable introductions (be it a book, an article, lectures notes, etc.) to compactly generated spaces that provides working knowledge for use in algebraic topology?
Thanks for any advice!
algebraic-topology soft-question
asked Dec 17 '18 at 13:57
ColescuColescu
3,26211136
$endgroup$
add a comment |
$begingroup$
In algebraic topology, many point-set topology problems arise. For example, the product of quotient maps need not be a quotient map; the smash product may not be associative; the canonical map $Z^{Xtimes Y}cong(Z^Y)^X$ is not necessarily a homeomorphism...
It seems that many authors chooses to stay in the category of compactly generated (weak Hausdorff) spaces to remedy this. For example, in Chapter 5 of Algebraic Topology by J. P. May, he introduces this concept and states without proof some basic properties, and then he assumes all topological spaces are compactly generated in the remainder of the book.
On the other hand, some authors do not assume this. Then some restrictions are necessary. In this context local compactness frequently crops up.
Here are my questions:
- Which approach should I take, as a beginner in this subject who does not want to be overwhelmed by technicalities?
- I currently know nothing about compactly generated spaces. I really want to read the text by May, but he uses this throughout, making many of his assertions simply false for me without the necessary restrictions (for instance, a cofibration need not be closed). How should I deal with this problem?
- Are there any readable introductions (be it a book, an article, lectures notes, etc.) to compactly generated spaces that provides working knowledge for use in algebraic topology?
Thanks for any advice!
algebraic-topology soft-question
asked Dec 17 '18 at 13:57
ColescuColescu
3,26211136
$endgroup$
$begingroup$
Very carefully. (This snarky comment means that I have absolutely no idea how to answer, or even understand, these questions.)
$endgroup$
– marty cohen
Dec 17 '18 at 14:22
2
$begingroup$
This is a common problem and was a problem for me as a beginner many years ago. I have no good answer. People make the assumptions that make their theorems provable. You can start by reading Steenrod's "A Convenient Category..." paper.
$endgroup$
– Randall
Dec 17 '18 at 14:24
2
$begingroup$
And if you are talking about JP May's A Concise Course in Algebraic Topology, that is an excellent book but not a great one to learn from: it's written in a sophisticated, non-basic manner. If you're just starting out, there are easier ways to learn. (With all due respect to JPM, who is a superb writer and my mathematical grandfather.)
$endgroup$
– Randall
Dec 17 '18 at 14:35
1
$begingroup$
If you care mostly about "algebraic" in algebraic topology, stick with polyhedra. I can see that you don't but that's your own choice.
$endgroup$
– Peter Saveliev
Dec 17 '18 at 16:57
$begingroup$
Steenrod's notes were already mentioned. I also commend Stricklands The Category of CGWH Spaces
$endgroup$
– Card_Trick
Dec 18 '18 at 21:58
add a comment |
$begingroup$
In algebraic topology, many point-set topology problems arise. For example, the product of quotient maps need not be a quotient map; the smash product may not be associative; the canonical map $Z^{Xtimes Y}cong(Z^Y)^X$ is not necessarily a homeomorphism...
It seems that many authors chooses to stay in the category of compactly generated (weak Hausdorff) spaces to remedy this. For example, in Chapter 5 of Algebraic Topology by J. P. May, he introduces this concept and states without proof some basic properties, and then he assumes all topological spaces are compactly generated in the remainder of the book.
On the other hand, some authors do not assume this. Then some restrictions are necessary. In this context local compactness frequently crops up.
Here are my questions:
- Which approach should I take, as a beginner in this subject who does not want to be overwhelmed by technicalities?
- I currently know nothing about compactly generated spaces. I really want to read the text by May, but he uses this throughout, making many of his assertions simply false for me without the necessary restrictions (for instance, a cofibration need not be closed). How should I deal with this problem?
- Are there any readable introductions (be it a book, an article, lectures notes, etc.) to compactly generated spaces that provides working knowledge for use in algebraic topology?
Thanks for any advice!
algebraic-topology soft-question
asked Dec 17 '18 at 13:57
ColescuColescu
3,26211136
$endgroup$
In algebraic topology, many point-set topology problems arise. For example, the product of quotient maps need not be a quotient map; the smash product may not be associative; the canonical map $Z^{Xtimes Y}cong(Z^Y)^X$ is not necessarily a homeomorphism...
It seems that many authors chooses to stay in the category of compactly generated (weak Hausdorff) spaces to remedy this. For example, in Chapter 5 of Algebraic Topology by J. P. May, he introduces this concept and states without proof some basic properties, and then he assumes all topological spaces are compactly generated in the remainder of the book.
On the other hand, some authors do not assume this. Then some restrictions are necessary. In this context local compactness frequently crops up.
Here are my questions:
- Which approach should I take, as a beginner in this subject who does not want to be overwhelmed by technicalities?
- I currently know nothing about compactly generated spaces. I really want to read the text by May, but he uses this throughout, making many of his assertions simply false for me without the necessary restrictions (for instance, a cofibration need not be closed). How should I deal with this problem?
- Are there any readable introductions (be it a book, an article, lectures notes, etc.) to compactly generated spaces that provides working knowledge for use in algebraic topology?
Thanks for any advice!
algebraic-topology soft-question
algebraic-topology soft-question
asked Dec 17 '18 at 13:57
ColescuColescu
3,26211136
asked Dec 17 '18 at 13:57
ColescuColescu
3,26211136
asked Dec 17 '18 at 13:57
ColescuColescu
3,26211136
asked Dec 17 '18 at 13:57
ColescuColescu
3,26211136
asked Dec 17 '18 at 13:57
ColescuColescu
3,26211136
3,26211136
$begingroup$
Very carefully. (This snarky comment means that I have absolutely no idea how to answer, or even understand, these questions.)
$endgroup$
– marty cohen
Dec 17 '18 at 14:22
2
$begingroup$
This is a common problem and was a problem for me as a beginner many years ago. I have no good answer. People make the assumptions that make their theorems provable. You can start by reading Steenrod's "A Convenient Category..." paper.
$endgroup$
– Randall
Dec 17 '18 at 14:24
2
$begingroup$
And if you are talking about JP May's A Concise Course in Algebraic Topology, that is an excellent book but not a great one to learn from: it's written in a sophisticated, non-basic manner. If you're just starting out, there are easier ways to learn. (With all due respect to JPM, who is a superb writer and my mathematical grandfather.)
$endgroup$
– Randall
Dec 17 '18 at 14:35
1
$begingroup$
If you care mostly about "algebraic" in algebraic topology, stick with polyhedra. I can see that you don't but that's your own choice.
$endgroup$
– Peter Saveliev
Dec 17 '18 at 16:57
$begingroup$
Steenrod's notes were already mentioned. I also commend Stricklands The Category of CGWH Spaces
$endgroup$
– Card_Trick
Dec 18 '18 at 21:58
add a comment |
$begingroup$
Very carefully. (This snarky comment means that I have absolutely no idea how to answer, or even understand, these questions.)
$endgroup$
– marty cohen
Dec 17 '18 at 14:22
2
$begingroup$
This is a common problem and was a problem for me as a beginner many years ago. I have no good answer. People make the assumptions that make their theorems provable. You can start by reading Steenrod's "A Convenient Category..." paper.
$endgroup$
– Randall
Dec 17 '18 at 14:24
2
$begingroup$
And if you are talking about JP May's A Concise Course in Algebraic Topology, that is an excellent book but not a great one to learn from: it's written in a sophisticated, non-basic manner. If you're just starting out, there are easier ways to learn. (With all due respect to JPM, who is a superb writer and my mathematical grandfather.)
$endgroup$
– Randall
Dec 17 '18 at 14:35
1
$begingroup$
If you care mostly about "algebraic" in algebraic topology, stick with polyhedra. I can see that you don't but that's your own choice.
$endgroup$
– Peter Saveliev
Dec 17 '18 at 16:57
$begingroup$
Steenrod's notes were already mentioned. I also commend Stricklands The Category of CGWH Spaces
$endgroup$
– Card_Trick
Dec 18 '18 at 21:58
$begingroup$
Very carefully. (This snarky comment means that I have absolutely no idea how to answer, or even understand, these questions.)
$endgroup$
– marty cohen
Dec 17 '18 at 14:22
$begingroup$
Very carefully. (This snarky comment means that I have absolutely no idea how to answer, or even understand, these questions.)
$endgroup$
– marty cohen
Dec 17 '18 at 14:22
2
2
$begingroup$
This is a common problem and was a problem for me as a beginner many years ago. I have no good answer. People make the assumptions that make their theorems provable. You can start by reading Steenrod's "A Convenient Category..." paper.
$endgroup$
– Randall
Dec 17 '18 at 14:24
$begingroup$
This is a common problem and was a problem for me as a beginner many years ago. I have no good answer. People make the assumptions that make their theorems provable. You can start by reading Steenrod's "A Convenient Category..." paper.
$endgroup$
– Randall
Dec 17 '18 at 14:24
2
2
$begingroup$
And if you are talking about JP May's A Concise Course in Algebraic Topology, that is an excellent book but not a great one to learn from: it's written in a sophisticated, non-basic manner. If you're just starting out, there are easier ways to learn. (With all due respect to JPM, who is a superb writer and my mathematical grandfather.)
$endgroup$
– Randall
Dec 17 '18 at 14:35
$begingroup$
And if you are talking about JP May's A Concise Course in Algebraic Topology, that is an excellent book but not a great one to learn from: it's written in a sophisticated, non-basic manner. If you're just starting out, there are easier ways to learn. (With all due respect to JPM, who is a superb writer and my mathematical grandfather.)
$endgroup$
– Randall
Dec 17 '18 at 14:35
1
1
$begingroup$
If you care mostly about "algebraic" in algebraic topology, stick with polyhedra. I can see that you don't but that's your own choice.
$endgroup$
– Peter Saveliev
Dec 17 '18 at 16:57
$begingroup$
If you care mostly about "algebraic" in algebraic topology, stick with polyhedra. I can see that you don't but that's your own choice.
$endgroup$
– Peter Saveliev
Dec 17 '18 at 16:57
$begingroup$
Steenrod's notes were already mentioned. I also commend Stricklands The Category of CGWH Spaces
$endgroup$
– Card_Trick
Dec 18 '18 at 21:58
$begingroup$
Steenrod's notes were already mentioned. I also commend Stricklands The Category of CGWH Spaces
$endgroup$
– Card_Trick
Dec 18 '18 at 21:58
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Some thoughts:
You should learn to cope with different assumptions. This will take time; I'm sorry. You will not learn all this overnight. You will commonly come across CW restrictions, simplicial stuff, and the compactly-generated category. There is really no way around this, since all these categories have their own intrinsic advantages. Ease with these notions comes only with experience using them.
See point 1. When you need to, step aside and prove what he says is obvious. Or, just assume it's all fine and keep reading. Sweat the technicalities when you need to, not when someone else does.
Steenrod's paper A Convenient Category of Topological Spaces was one of the first to advocate for compactly-generated spaces. It actually contains proofs of many of the things you're asking about.
Michigan Math. J., Volume 14, Issue 2 (1967), 133-152.
answered Dec 17 '18 at 14:30
RandallRandall
10.6k11431
$endgroup$
$begingroup$
Steenrod's paper has some queries about it: see the history in ncatlab.org/nlab/show/convenient+category+of+topological+spaces I hope my remarks in my book "Topology and Groupoids" get the message over that the search for a category "adequate and convenient for all purposes of topology" still has a way to go!
$endgroup$
– Ronnie Brown
Dec 17 '18 at 14:54
$begingroup$
Thank you for your insightful comments! I guess maybe I should just try to prove eveything May says then. My doubt is basically this: Not every author uses compactly generated spaces systematically. So would it be better if (as a beginner) I choose to stay in the usual category of topological spaces? I'll definitely look into that paper you've mentioned!
$endgroup$
– Colescu
Dec 17 '18 at 15:00
$begingroup$
I think Munkres' book Elements Of Algebraic Topology has careful proofs of the basic results in the standard and weak topologies, with good examples to show the differences. He starts early with Simplicial Complexes and then when CW complexes re considered, goes into more detail.
$endgroup$
– Matematleta
Dec 17 '18 at 16:33
add a comment |
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$begingroup$
Some thoughts:
You should learn to cope with different assumptions. This will take time; I'm sorry. You will not learn all this overnight. You will commonly come across CW restrictions, simplicial stuff, and the compactly-generated category. There is really no way around this, since all these categories have their own intrinsic advantages. Ease with these notions comes only with experience using them.
See point 1. When you need to, step aside and prove what he says is obvious. Or, just assume it's all fine and keep reading. Sweat the technicalities when you need to, not when someone else does.
Steenrod's paper A Convenient Category of Topological Spaces was one of the first to advocate for compactly-generated spaces. It actually contains proofs of many of the things you're asking about.
Michigan Math. J., Volume 14, Issue 2 (1967), 133-152.
answered Dec 17 '18 at 14:30
RandallRandall
10.6k11431
$endgroup$
$begingroup$
Steenrod's paper has some queries about it: see the history in ncatlab.org/nlab/show/convenient+category+of+topological+spaces I hope my remarks in my book "Topology and Groupoids" get the message over that the search for a category "adequate and convenient for all purposes of topology" still has a way to go!
$endgroup$
– Ronnie Brown
Dec 17 '18 at 14:54
$begingroup$
Thank you for your insightful comments! I guess maybe I should just try to prove eveything May says then. My doubt is basically this: Not every author uses compactly generated spaces systematically. So would it be better if (as a beginner) I choose to stay in the usual category of topological spaces? I'll definitely look into that paper you've mentioned!
$endgroup$
– Colescu
Dec 17 '18 at 15:00
$begingroup$
I think Munkres' book Elements Of Algebraic Topology has careful proofs of the basic results in the standard and weak topologies, with good examples to show the differences. He starts early with Simplicial Complexes and then when CW complexes re considered, goes into more detail.
$endgroup$
– Matematleta
Dec 17 '18 at 16:33
add a comment |
$begingroup$
Some thoughts:
You should learn to cope with different assumptions. This will take time; I'm sorry. You will not learn all this overnight. You will commonly come across CW restrictions, simplicial stuff, and the compactly-generated category. There is really no way around this, since all these categories have their own intrinsic advantages. Ease with these notions comes only with experience using them.
See point 1. When you need to, step aside and prove what he says is obvious. Or, just assume it's all fine and keep reading. Sweat the technicalities when you need to, not when someone else does.
Steenrod's paper A Convenient Category of Topological Spaces was one of the first to advocate for compactly-generated spaces. It actually contains proofs of many of the things you're asking about.
Michigan Math. J., Volume 14, Issue 2 (1967), 133-152.
answered Dec 17 '18 at 14:30
RandallRandall
10.6k11431
$endgroup$
$begingroup$
Steenrod's paper has some queries about it: see the history in ncatlab.org/nlab/show/convenient+category+of+topological+spaces I hope my remarks in my book "Topology and Groupoids" get the message over that the search for a category "adequate and convenient for all purposes of topology" still has a way to go!
$endgroup$
– Ronnie Brown
Dec 17 '18 at 14:54
$begingroup$
Thank you for your insightful comments! I guess maybe I should just try to prove eveything May says then. My doubt is basically this: Not every author uses compactly generated spaces systematically. So would it be better if (as a beginner) I choose to stay in the usual category of topological spaces? I'll definitely look into that paper you've mentioned!
$endgroup$
– Colescu
Dec 17 '18 at 15:00
$begingroup$
I think Munkres' book Elements Of Algebraic Topology has careful proofs of the basic results in the standard and weak topologies, with good examples to show the differences. He starts early with Simplicial Complexes and then when CW complexes re considered, goes into more detail.
$endgroup$
– Matematleta
Dec 17 '18 at 16:33
add a comment |
$begingroup$
Some thoughts:
You should learn to cope with different assumptions. This will take time; I'm sorry. You will not learn all this overnight. You will commonly come across CW restrictions, simplicial stuff, and the compactly-generated category. There is really no way around this, since all these categories have their own intrinsic advantages. Ease with these notions comes only with experience using them.
See point 1. When you need to, step aside and prove what he says is obvious. Or, just assume it's all fine and keep reading. Sweat the technicalities when you need to, not when someone else does.
Steenrod's paper A Convenient Category of Topological Spaces was one of the first to advocate for compactly-generated spaces. It actually contains proofs of many of the things you're asking about.
Michigan Math. J., Volume 14, Issue 2 (1967), 133-152.
answered Dec 17 '18 at 14:30
RandallRandall
10.6k11431
$endgroup$
Some thoughts:
You should learn to cope with different assumptions. This will take time; I'm sorry. You will not learn all this overnight. You will commonly come across CW restrictions, simplicial stuff, and the compactly-generated category. There is really no way around this, since all these categories have their own intrinsic advantages. Ease with these notions comes only with experience using them.
See point 1. When you need to, step aside and prove what he says is obvious. Or, just assume it's all fine and keep reading. Sweat the technicalities when you need to, not when someone else does.
Steenrod's paper A Convenient Category of Topological Spaces was one of the first to advocate for compactly-generated spaces. It actually contains proofs of many of the things you're asking about.
Michigan Math. J., Volume 14, Issue 2 (1967), 133-152.
answered Dec 17 '18 at 14:30
RandallRandall
10.6k11431
answered Dec 17 '18 at 14:30
RandallRandall
10.6k11431
answered Dec 17 '18 at 14:30
RandallRandall
10.6k11431
answered Dec 17 '18 at 14:30
RandallRandall
10.6k11431
10.6k11431
$begingroup$
Steenrod's paper has some queries about it: see the history in ncatlab.org/nlab/show/convenient+category+of+topological+spaces I hope my remarks in my book "Topology and Groupoids" get the message over that the search for a category "adequate and convenient for all purposes of topology" still has a way to go!
$endgroup$
– Ronnie Brown
Dec 17 '18 at 14:54
$begingroup$
Thank you for your insightful comments! I guess maybe I should just try to prove eveything May says then. My doubt is basically this: Not every author uses compactly generated spaces systematically. So would it be better if (as a beginner) I choose to stay in the usual category of topological spaces? I'll definitely look into that paper you've mentioned!
$endgroup$
– Colescu
Dec 17 '18 at 15:00
$begingroup$
I think Munkres' book Elements Of Algebraic Topology has careful proofs of the basic results in the standard and weak topologies, with good examples to show the differences. He starts early with Simplicial Complexes and then when CW complexes re considered, goes into more detail.
$endgroup$
– Matematleta
Dec 17 '18 at 16:33
add a comment |
$begingroup$
Steenrod's paper has some queries about it: see the history in ncatlab.org/nlab/show/convenient+category+of+topological+spaces I hope my remarks in my book "Topology and Groupoids" get the message over that the search for a category "adequate and convenient for all purposes of topology" still has a way to go!
$endgroup$
– Ronnie Brown
Dec 17 '18 at 14:54
$begingroup$
Thank you for your insightful comments! I guess maybe I should just try to prove eveything May says then. My doubt is basically this: Not every author uses compactly generated spaces systematically. So would it be better if (as a beginner) I choose to stay in the usual category of topological spaces? I'll definitely look into that paper you've mentioned!
$endgroup$
– Colescu
Dec 17 '18 at 15:00
$begingroup$
I think Munkres' book Elements Of Algebraic Topology has careful proofs of the basic results in the standard and weak topologies, with good examples to show the differences. He starts early with Simplicial Complexes and then when CW complexes re considered, goes into more detail.
$endgroup$
– Matematleta
Dec 17 '18 at 16:33
$begingroup$
Steenrod's paper has some queries about it: see the history in ncatlab.org/nlab/show/convenient+category+of+topological+spaces I hope my remarks in my book "Topology and Groupoids" get the message over that the search for a category "adequate and convenient for all purposes of topology" still has a way to go!
$endgroup$
– Ronnie Brown
Dec 17 '18 at 14:54
$begingroup$
Steenrod's paper has some queries about it: see the history in ncatlab.org/nlab/show/convenient+category+of+topological+spaces I hope my remarks in my book "Topology and Groupoids" get the message over that the search for a category "adequate and convenient for all purposes of topology" still has a way to go!
$endgroup$
– Ronnie Brown
Dec 17 '18 at 14:54
$begingroup$
Thank you for your insightful comments! I guess maybe I should just try to prove eveything May says then. My doubt is basically this: Not every author uses compactly generated spaces systematically. So would it be better if (as a beginner) I choose to stay in the usual category of topological spaces? I'll definitely look into that paper you've mentioned!
$endgroup$
– Colescu
Dec 17 '18 at 15:00
$begingroup$
Thank you for your insightful comments! I guess maybe I should just try to prove eveything May says then. My doubt is basically this: Not every author uses compactly generated spaces systematically. So would it be better if (as a beginner) I choose to stay in the usual category of topological spaces? I'll definitely look into that paper you've mentioned!
$endgroup$
– Colescu
Dec 17 '18 at 15:00
$begingroup$
I think Munkres' book Elements Of Algebraic Topology has careful proofs of the basic results in the standard and weak topologies, with good examples to show the differences. He starts early with Simplicial Complexes and then when CW complexes re considered, goes into more detail.
$endgroup$
– Matematleta
Dec 17 '18 at 16:33
$begingroup$
I think Munkres' book Elements Of Algebraic Topology has careful proofs of the basic results in the standard and weak topologies, with good examples to show the differences. He starts early with Simplicial Complexes and then when CW complexes re considered, goes into more detail.
$endgroup$
– Matematleta
Dec 17 '18 at 16:33
add a comment |
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Very carefully. (This snarky comment means that I have absolutely no idea how to answer, or even understand, these questions.)
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– marty cohen
Dec 17 '18 at 14:22
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This is a common problem and was a problem for me as a beginner many years ago. I have no good answer. People make the assumptions that make their theorems provable. You can start by reading Steenrod's "A Convenient Category..." paper.
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– Randall
Dec 17 '18 at 14:24
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And if you are talking about JP May's A Concise Course in Algebraic Topology, that is an excellent book but not a great one to learn from: it's written in a sophisticated, non-basic manner. If you're just starting out, there are easier ways to learn. (With all due respect to JPM, who is a superb writer and my mathematical grandfather.)
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– Randall
Dec 17 '18 at 14:35
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If you care mostly about "algebraic" in algebraic topology, stick with polyhedra. I can see that you don't but that's your own choice.
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– Peter Saveliev
Dec 17 '18 at 16:57
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Steenrod's notes were already mentioned. I also commend Stricklands The Category of CGWH Spaces
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– Card_Trick
Dec 18 '18 at 21:58