A map of non-pathological topology?What do absolute neighborhood retracts look like?Which is the correct ring of functions for a topological space?Various concepts of “closure” or “completion” in mathematicsWhy should algebraic objects have naturally associated topological spaces? (Formerly: What is a topological space?)How should one think about non-Hausdorff topologies?Questions about spectra of rings of continuous functionsWhy chain homotopy when there is no topology in the background?Status of PL topologyCompact subsets and Hausdorffness of TopologyName of the concept “Topological boundary of A intersected with A”Local “pathologies” in spaces arising naturally in algebraic topology
A map of non-pathological topology?
What do absolute neighborhood retracts look like?Which is the correct ring of functions for a topological space?Various concepts of “closure” or “completion” in mathematicsWhy should algebraic objects have naturally associated topological spaces? (Formerly: What is a topological space?)How should one think about non-Hausdorff topologies?Questions about spectra of rings of continuous functionsWhy chain homotopy when there is no topology in the background?Status of PL topologyCompact subsets and Hausdorffness of TopologyName of the concept “Topological boundary of A intersected with A”Local “pathologies” in spaces arising naturally in algebraic topology
$begingroup$
I think of topological spaces as coming in several "islands of interestingness" (the CW island, the Zariski archipelago,...) dotting a vast "pathological sea" (the long line ocean, the gulf of the lower limit...). That is, I only know how to think about a topological space if it happens to live on one of these islands, the methods appropriate to one island may be completely unrelated to those of another, and a "random" topological space is probably unrelated to anything I know how to think about and is thus "pathological". I'd like to get a better perspective on how many of these islands there are -- and perhaps whether some which I think are distinct are actually connected by some isthmus. Here's what my current map looks like:
CW complexes (and spaces homotopy equivalent to such)
Zariski spectra of commutative rings (and schemes)
Stone spaces (totally disconnected compact Hausdorff spaces)
Infinite-dimensional topological vector spaces (and spaces locally modeled on them)
One of the characteristic features of this map is that there is little overlap between the islands although there is overlap between these islands in a literal sense, what really sets them apart is that the tools used in exploring one island bear little resemblance to those used for another. For example, when studying spaces using CW complex tools, non-Hausdorffness is regarded as pathological, clopen sets as uninteresting, and infinite-dimensionality as an annoyance whereas such features are respectively embraced when studying Zariski spectra, Stone spaces, and functional-analytic spaces.
Questions:
Are there other classes of topological spaces which are interesting to study (and not just as a source of pathologies)?
Are these islands less isolated than I'm making them out to be? E.g. are there interesting topological considerations to be made which apply simultaneously to, say, Banach spaces and Stone spaces?
Is it correct to think that the ocean is vast, i.e. that "most" topological spaces are "pathological"?
gn.general-topology soft-question big-picture
asked Jun 6 at 4:57
Tim CampionTim Campion
17.6k5 gold badges62 silver badges151 bronze badges
$endgroup$
|
show 16 more comments
$begingroup$
I think of topological spaces as coming in several "islands of interestingness" (the CW island, the Zariski archipelago,...) dotting a vast "pathological sea" (the long line ocean, the gulf of the lower limit...). That is, I only know how to think about a topological space if it happens to live on one of these islands, the methods appropriate to one island may be completely unrelated to those of another, and a "random" topological space is probably unrelated to anything I know how to think about and is thus "pathological". I'd like to get a better perspective on how many of these islands there are -- and perhaps whether some which I think are distinct are actually connected by some isthmus. Here's what my current map looks like:
CW complexes (and spaces homotopy equivalent to such)
Zariski spectra of commutative rings (and schemes)
Stone spaces (totally disconnected compact Hausdorff spaces)
Infinite-dimensional topological vector spaces (and spaces locally modeled on them)
One of the characteristic features of this map is that there is little overlap between the islands although there is overlap between these islands in a literal sense, what really sets them apart is that the tools used in exploring one island bear little resemblance to those used for another. For example, when studying spaces using CW complex tools, non-Hausdorffness is regarded as pathological, clopen sets as uninteresting, and infinite-dimensionality as an annoyance whereas such features are respectively embraced when studying Zariski spectra, Stone spaces, and functional-analytic spaces.
Questions:
Are there other classes of topological spaces which are interesting to study (and not just as a source of pathologies)?
Are these islands less isolated than I'm making them out to be? E.g. are there interesting topological considerations to be made which apply simultaneously to, say, Banach spaces and Stone spaces?
Is it correct to think that the ocean is vast, i.e. that "most" topological spaces are "pathological"?
gn.general-topology soft-question big-picture
asked Jun 6 at 4:57
Tim CampionTim Campion
17.6k5 gold badges62 silver badges151 bronze badges
$endgroup$
8
$begingroup$
Stone spaces are Zariski spectra of Boolean rings.
$endgroup$
– Todd Trimble♦
Jun 6 at 5:00
9
$begingroup$
Polish spaces as another class?
$endgroup$
– Todd Trimble♦
Jun 6 at 5:17
7
$begingroup$
Another vote for Polish spaces from me. They're the typical "nice" space for measure theory, probability, and descriptive set theory. Elsewhere in analysis, "locally compact Hausdorff" is an important island.
$endgroup$
– Nate Eldredge
Jun 6 at 6:32
8
$begingroup$
The sea of pathology has various levels of depth: there is the deep sea of non-$T_0$ spaces, but there are deeper trenches: pretopological spaces and pseudotopological spaces. Of what unseen horrors in these depths may lurk few have lived to tell the tale. Also, let's not forget the great ocean of locales (connected by the strait of sober spaces) in which perhaps more islands of non-pathology can be found.
$endgroup$
– Gro-Tsen
Jun 6 at 12:00
5
$begingroup$
I find it difficult to see how $omega_1+1$ with its order topology is on an island (of Stone spaces) whereas inserting pieces of $mathbb R$ to make the long line pushes the space into the ocean.
$endgroup$
– Andreas Blass
Jun 6 at 17:01
|
show 16 more comments
$begingroup$
I think of topological spaces as coming in several "islands of interestingness" (the CW island, the Zariski archipelago,...) dotting a vast "pathological sea" (the long line ocean, the gulf of the lower limit...). That is, I only know how to think about a topological space if it happens to live on one of these islands, the methods appropriate to one island may be completely unrelated to those of another, and a "random" topological space is probably unrelated to anything I know how to think about and is thus "pathological". I'd like to get a better perspective on how many of these islands there are -- and perhaps whether some which I think are distinct are actually connected by some isthmus. Here's what my current map looks like:
CW complexes (and spaces homotopy equivalent to such)
Zariski spectra of commutative rings (and schemes)
Stone spaces (totally disconnected compact Hausdorff spaces)
Infinite-dimensional topological vector spaces (and spaces locally modeled on them)
One of the characteristic features of this map is that there is little overlap between the islands although there is overlap between these islands in a literal sense, what really sets them apart is that the tools used in exploring one island bear little resemblance to those used for another. For example, when studying spaces using CW complex tools, non-Hausdorffness is regarded as pathological, clopen sets as uninteresting, and infinite-dimensionality as an annoyance whereas such features are respectively embraced when studying Zariski spectra, Stone spaces, and functional-analytic spaces.
Questions:
Are there other classes of topological spaces which are interesting to study (and not just as a source of pathologies)?
Are these islands less isolated than I'm making them out to be? E.g. are there interesting topological considerations to be made which apply simultaneously to, say, Banach spaces and Stone spaces?
Is it correct to think that the ocean is vast, i.e. that "most" topological spaces are "pathological"?
gn.general-topology soft-question big-picture
asked Jun 6 at 4:57
Tim CampionTim Campion
17.6k5 gold badges62 silver badges151 bronze badges
$endgroup$
I think of topological spaces as coming in several "islands of interestingness" (the CW island, the Zariski archipelago,...) dotting a vast "pathological sea" (the long line ocean, the gulf of the lower limit...). That is, I only know how to think about a topological space if it happens to live on one of these islands, the methods appropriate to one island may be completely unrelated to those of another, and a "random" topological space is probably unrelated to anything I know how to think about and is thus "pathological". I'd like to get a better perspective on how many of these islands there are -- and perhaps whether some which I think are distinct are actually connected by some isthmus. Here's what my current map looks like:
CW complexes (and spaces homotopy equivalent to such)
Zariski spectra of commutative rings (and schemes)
Stone spaces (totally disconnected compact Hausdorff spaces)
Infinite-dimensional topological vector spaces (and spaces locally modeled on them)
One of the characteristic features of this map is that there is little overlap between the islands although there is overlap between these islands in a literal sense, what really sets them apart is that the tools used in exploring one island bear little resemblance to those used for another. For example, when studying spaces using CW complex tools, non-Hausdorffness is regarded as pathological, clopen sets as uninteresting, and infinite-dimensionality as an annoyance whereas such features are respectively embraced when studying Zariski spectra, Stone spaces, and functional-analytic spaces.
Questions:
Are there other classes of topological spaces which are interesting to study (and not just as a source of pathologies)?
Are these islands less isolated than I'm making them out to be? E.g. are there interesting topological considerations to be made which apply simultaneously to, say, Banach spaces and Stone spaces?
Is it correct to think that the ocean is vast, i.e. that "most" topological spaces are "pathological"?
gn.general-topology soft-question big-picture
gn.general-topology soft-question big-picture
asked Jun 6 at 4:57
Tim CampionTim Campion
17.6k5 gold badges62 silver badges151 bronze badges
asked Jun 6 at 4:57
Tim CampionTim Campion
17.6k5 gold badges62 silver badges151 bronze badges
edited Jun 6 at 5:21
Tim Campion
asked Jun 6 at 4:57
Tim CampionTim Campion
17.6k5 gold badges62 silver badges151 bronze badges
asked Jun 6 at 4:57
Tim CampionTim Campion
17.6k5 gold badges62 silver badges151 bronze badges
asked Jun 6 at 4:57
Tim CampionTim Campion
17.6k5 gold badges62 silver badges151 bronze badges
17.6k5 gold badges62 silver badges151 bronze badges
8
$begingroup$
Stone spaces are Zariski spectra of Boolean rings.
$endgroup$
– Todd Trimble♦
Jun 6 at 5:00
9
$begingroup$
Polish spaces as another class?
$endgroup$
– Todd Trimble♦
Jun 6 at 5:17
7
$begingroup$
Another vote for Polish spaces from me. They're the typical "nice" space for measure theory, probability, and descriptive set theory. Elsewhere in analysis, "locally compact Hausdorff" is an important island.
$endgroup$
– Nate Eldredge
Jun 6 at 6:32
8
$begingroup$
The sea of pathology has various levels of depth: there is the deep sea of non-$T_0$ spaces, but there are deeper trenches: pretopological spaces and pseudotopological spaces. Of what unseen horrors in these depths may lurk few have lived to tell the tale. Also, let's not forget the great ocean of locales (connected by the strait of sober spaces) in which perhaps more islands of non-pathology can be found.
$endgroup$
– Gro-Tsen
Jun 6 at 12:00
5
$begingroup$
I find it difficult to see how $omega_1+1$ with its order topology is on an island (of Stone spaces) whereas inserting pieces of $mathbb R$ to make the long line pushes the space into the ocean.
$endgroup$
– Andreas Blass
Jun 6 at 17:01
|
show 16 more comments
8
$begingroup$
Stone spaces are Zariski spectra of Boolean rings.
$endgroup$
– Todd Trimble♦
Jun 6 at 5:00
9
$begingroup$
Polish spaces as another class?
$endgroup$
– Todd Trimble♦
Jun 6 at 5:17
7
$begingroup$
Another vote for Polish spaces from me. They're the typical "nice" space for measure theory, probability, and descriptive set theory. Elsewhere in analysis, "locally compact Hausdorff" is an important island.
$endgroup$
– Nate Eldredge
Jun 6 at 6:32
8
$begingroup$
The sea of pathology has various levels of depth: there is the deep sea of non-$T_0$ spaces, but there are deeper trenches: pretopological spaces and pseudotopological spaces. Of what unseen horrors in these depths may lurk few have lived to tell the tale. Also, let's not forget the great ocean of locales (connected by the strait of sober spaces) in which perhaps more islands of non-pathology can be found.
$endgroup$
– Gro-Tsen
Jun 6 at 12:00
5
$begingroup$
I find it difficult to see how $omega_1+1$ with its order topology is on an island (of Stone spaces) whereas inserting pieces of $mathbb R$ to make the long line pushes the space into the ocean.
$endgroup$
– Andreas Blass
Jun 6 at 17:01
8
8
$begingroup$
Stone spaces are Zariski spectra of Boolean rings.
$endgroup$
– Todd Trimble♦
Jun 6 at 5:00
$begingroup$
Stone spaces are Zariski spectra of Boolean rings.
$endgroup$
– Todd Trimble♦
Jun 6 at 5:00
9
9
$begingroup$
Polish spaces as another class?
$endgroup$
– Todd Trimble♦
Jun 6 at 5:17
$begingroup$
Polish spaces as another class?
$endgroup$
– Todd Trimble♦
Jun 6 at 5:17
7
7
$begingroup$
Another vote for Polish spaces from me. They're the typical "nice" space for measure theory, probability, and descriptive set theory. Elsewhere in analysis, "locally compact Hausdorff" is an important island.
$endgroup$
– Nate Eldredge
Jun 6 at 6:32
$begingroup$
Another vote for Polish spaces from me. They're the typical "nice" space for measure theory, probability, and descriptive set theory. Elsewhere in analysis, "locally compact Hausdorff" is an important island.
$endgroup$
– Nate Eldredge
Jun 6 at 6:32
8
8
$begingroup$
The sea of pathology has various levels of depth: there is the deep sea of non-$T_0$ spaces, but there are deeper trenches: pretopological spaces and pseudotopological spaces. Of what unseen horrors in these depths may lurk few have lived to tell the tale. Also, let's not forget the great ocean of locales (connected by the strait of sober spaces) in which perhaps more islands of non-pathology can be found.
$endgroup$
– Gro-Tsen
Jun 6 at 12:00
$begingroup$
The sea of pathology has various levels of depth: there is the deep sea of non-$T_0$ spaces, but there are deeper trenches: pretopological spaces and pseudotopological spaces. Of what unseen horrors in these depths may lurk few have lived to tell the tale. Also, let's not forget the great ocean of locales (connected by the strait of sober spaces) in which perhaps more islands of non-pathology can be found.
$endgroup$
– Gro-Tsen
Jun 6 at 12:00
5
5
$begingroup$
I find it difficult to see how $omega_1+1$ with its order topology is on an island (of Stone spaces) whereas inserting pieces of $mathbb R$ to make the long line pushes the space into the ocean.
$endgroup$
– Andreas Blass
Jun 6 at 17:01
$begingroup$
I find it difficult to see how $omega_1+1$ with its order topology is on an island (of Stone spaces) whereas inserting pieces of $mathbb R$ to make the long line pushes the space into the ocean.
$endgroup$
– Andreas Blass
Jun 6 at 17:01
|
show 16 more comments
3 Answers
3
active
oldest
votes
$begingroup$
What about finite topological spaces? (A useful source of stuff on these is: Algebraic Topology of Finite Topological Spaces and Applications by Jonathan Barmak.) That area studies non-Hausdorff spaces most of the time and has strong links with CW-complexes via face posets but also via the link with posets has external contacts to combinatorics and to some of your other islands.
In another direction the use of topological spaces in Logic and Theoretical Computer Science should fit somewhere. One entry point is `Topology via Logic' by Steve Vickers. This fits near to some of your existing islands so will be linked to them by bridges (probably with tolls!). There is also a use of topological spaces within Modal Logic which again looks to be distinct to the others but linked.
Finally 'pathological' is not really definable except as meaning 'outside my current interests'! Pathology is in the eye of the beholder. Spaces such as compact Haudorff spaces have a decent algebraic topology if one uses strong shape theory. This approximates these spaces by CW spaces and transfers the well loved homotopy theory of those across using procategorical methods. Even general closed subsets of $mathbbR^n$ which can look pathological can be explored. There are connections between their $C^*$-algebras and their strong shape, so linking the Banach space approaches with an extended CW-approach.
(I will stop there as that leads off into non-commutative spaces, and lots of other lovely areas, such as sheaves and toposes, but is getting to the limits of stuff I know at all well!)
answered Jun 6 at 5:36
Tim PorterTim Porter
7,7321 gold badge18 silver badges34 bronze badges
$endgroup$
2
$begingroup$
Some of the 'islands' are very close to each other, so the analogy of a map has its limitations. Any compact metric (CM) space can be thought of as sitting in the Hilbert cube, and then it can be thought of as an intersection of polyhedral neighbourhoods so although CM spaces can be pathological from some points of view they are sort of infinitely near the polyhedral space island!!! It would be fun to try to extend the map with bridges between islands, (perhaps adjoint functors might serve for this) and then talk of reefs that are almost above sea level, etc.
$endgroup$
– Tim Porter
Jun 8 at 14:02
add a comment
|
$begingroup$
I'll go ahead and say that Polish spaces are an interesting and almost sui generis class. There is a rich literature of applications to and from descriptive set theory, with layers of "pathology" hierarchically organized along lines closely related to the arithmetical and analytic hierarchies. They are also widely used, as Nate mentioned, in abstract measure and probability theory.
I would say there are isthmuses connecting this class to the class of continua (mentioned by D.S. Lipham; consider for example the theory of pseudo-arcs) as compact connected metric spaces, and to some extent to locally compact Hausdorff spaces (e.g., a locally compact Hausdorff space is Polish iff it is second countable).
answered Jun 6 at 14:24
Todd Trimble♦Todd Trimble
44.6k5 gold badges158 silver badges270 bronze badges
$endgroup$
add a comment
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$begingroup$
o-minimal structures, as generalisation of “tame” topology of semi-algebraic and semi-analytic sets, are quite interesting, in applications in particular. See e.g. the book
https://books.google.co.uk/books/about/Tame_Topology_and_O_minimal_Structures.html
answered Jun 6 at 16:05
Dima PasechnikDima Pasechnik
10.6k2 gold badges22 silver badges57 bronze badges
$endgroup$
$begingroup$
A great book which deserves to be well-known.
$endgroup$
– Todd Trimble♦
Jun 8 at 11:49
add a comment
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
What about finite topological spaces? (A useful source of stuff on these is: Algebraic Topology of Finite Topological Spaces and Applications by Jonathan Barmak.) That area studies non-Hausdorff spaces most of the time and has strong links with CW-complexes via face posets but also via the link with posets has external contacts to combinatorics and to some of your other islands.
In another direction the use of topological spaces in Logic and Theoretical Computer Science should fit somewhere. One entry point is `Topology via Logic' by Steve Vickers. This fits near to some of your existing islands so will be linked to them by bridges (probably with tolls!). There is also a use of topological spaces within Modal Logic which again looks to be distinct to the others but linked.
Finally 'pathological' is not really definable except as meaning 'outside my current interests'! Pathology is in the eye of the beholder. Spaces such as compact Haudorff spaces have a decent algebraic topology if one uses strong shape theory. This approximates these spaces by CW spaces and transfers the well loved homotopy theory of those across using procategorical methods. Even general closed subsets of $mathbbR^n$ which can look pathological can be explored. There are connections between their $C^*$-algebras and their strong shape, so linking the Banach space approaches with an extended CW-approach.
(I will stop there as that leads off into non-commutative spaces, and lots of other lovely areas, such as sheaves and toposes, but is getting to the limits of stuff I know at all well!)
answered Jun 6 at 5:36
Tim PorterTim Porter
7,7321 gold badge18 silver badges34 bronze badges
$endgroup$
2
$begingroup$
Some of the 'islands' are very close to each other, so the analogy of a map has its limitations. Any compact metric (CM) space can be thought of as sitting in the Hilbert cube, and then it can be thought of as an intersection of polyhedral neighbourhoods so although CM spaces can be pathological from some points of view they are sort of infinitely near the polyhedral space island!!! It would be fun to try to extend the map with bridges between islands, (perhaps adjoint functors might serve for this) and then talk of reefs that are almost above sea level, etc.
$endgroup$
– Tim Porter
Jun 8 at 14:02
add a comment
|
$begingroup$
What about finite topological spaces? (A useful source of stuff on these is: Algebraic Topology of Finite Topological Spaces and Applications by Jonathan Barmak.) That area studies non-Hausdorff spaces most of the time and has strong links with CW-complexes via face posets but also via the link with posets has external contacts to combinatorics and to some of your other islands.
In another direction the use of topological spaces in Logic and Theoretical Computer Science should fit somewhere. One entry point is `Topology via Logic' by Steve Vickers. This fits near to some of your existing islands so will be linked to them by bridges (probably with tolls!). There is also a use of topological spaces within Modal Logic which again looks to be distinct to the others but linked.
Finally 'pathological' is not really definable except as meaning 'outside my current interests'! Pathology is in the eye of the beholder. Spaces such as compact Haudorff spaces have a decent algebraic topology if one uses strong shape theory. This approximates these spaces by CW spaces and transfers the well loved homotopy theory of those across using procategorical methods. Even general closed subsets of $mathbbR^n$ which can look pathological can be explored. There are connections between their $C^*$-algebras and their strong shape, so linking the Banach space approaches with an extended CW-approach.
(I will stop there as that leads off into non-commutative spaces, and lots of other lovely areas, such as sheaves and toposes, but is getting to the limits of stuff I know at all well!)
answered Jun 6 at 5:36
Tim PorterTim Porter
7,7321 gold badge18 silver badges34 bronze badges
$endgroup$
2
$begingroup$
Some of the 'islands' are very close to each other, so the analogy of a map has its limitations. Any compact metric (CM) space can be thought of as sitting in the Hilbert cube, and then it can be thought of as an intersection of polyhedral neighbourhoods so although CM spaces can be pathological from some points of view they are sort of infinitely near the polyhedral space island!!! It would be fun to try to extend the map with bridges between islands, (perhaps adjoint functors might serve for this) and then talk of reefs that are almost above sea level, etc.
$endgroup$
– Tim Porter
Jun 8 at 14:02
add a comment
|
$begingroup$
What about finite topological spaces? (A useful source of stuff on these is: Algebraic Topology of Finite Topological Spaces and Applications by Jonathan Barmak.) That area studies non-Hausdorff spaces most of the time and has strong links with CW-complexes via face posets but also via the link with posets has external contacts to combinatorics and to some of your other islands.
In another direction the use of topological spaces in Logic and Theoretical Computer Science should fit somewhere. One entry point is `Topology via Logic' by Steve Vickers. This fits near to some of your existing islands so will be linked to them by bridges (probably with tolls!). There is also a use of topological spaces within Modal Logic which again looks to be distinct to the others but linked.
Finally 'pathological' is not really definable except as meaning 'outside my current interests'! Pathology is in the eye of the beholder. Spaces such as compact Haudorff spaces have a decent algebraic topology if one uses strong shape theory. This approximates these spaces by CW spaces and transfers the well loved homotopy theory of those across using procategorical methods. Even general closed subsets of $mathbbR^n$ which can look pathological can be explored. There are connections between their $C^*$-algebras and their strong shape, so linking the Banach space approaches with an extended CW-approach.
(I will stop there as that leads off into non-commutative spaces, and lots of other lovely areas, such as sheaves and toposes, but is getting to the limits of stuff I know at all well!)
answered Jun 6 at 5:36
Tim PorterTim Porter
7,7321 gold badge18 silver badges34 bronze badges
$endgroup$
What about finite topological spaces? (A useful source of stuff on these is: Algebraic Topology of Finite Topological Spaces and Applications by Jonathan Barmak.) That area studies non-Hausdorff spaces most of the time and has strong links with CW-complexes via face posets but also via the link with posets has external contacts to combinatorics and to some of your other islands.
In another direction the use of topological spaces in Logic and Theoretical Computer Science should fit somewhere. One entry point is `Topology via Logic' by Steve Vickers. This fits near to some of your existing islands so will be linked to them by bridges (probably with tolls!). There is also a use of topological spaces within Modal Logic which again looks to be distinct to the others but linked.
Finally 'pathological' is not really definable except as meaning 'outside my current interests'! Pathology is in the eye of the beholder. Spaces such as compact Haudorff spaces have a decent algebraic topology if one uses strong shape theory. This approximates these spaces by CW spaces and transfers the well loved homotopy theory of those across using procategorical methods. Even general closed subsets of $mathbbR^n$ which can look pathological can be explored. There are connections between their $C^*$-algebras and their strong shape, so linking the Banach space approaches with an extended CW-approach.
(I will stop there as that leads off into non-commutative spaces, and lots of other lovely areas, such as sheaves and toposes, but is getting to the limits of stuff I know at all well!)
answered Jun 6 at 5:36
Tim PorterTim Porter
7,7321 gold badge18 silver badges34 bronze badges
edited Jun 8 at 9:21
answered Jun 6 at 5:36
Tim PorterTim Porter
7,7321 gold badge18 silver badges34 bronze badges
answered Jun 6 at 5:36
Tim PorterTim Porter
7,7321 gold badge18 silver badges34 bronze badges
answered Jun 6 at 5:36
Tim PorterTim Porter
7,7321 gold badge18 silver badges34 bronze badges
7,7321 gold badge18 silver badges34 bronze badges
2
$begingroup$
Some of the 'islands' are very close to each other, so the analogy of a map has its limitations. Any compact metric (CM) space can be thought of as sitting in the Hilbert cube, and then it can be thought of as an intersection of polyhedral neighbourhoods so although CM spaces can be pathological from some points of view they are sort of infinitely near the polyhedral space island!!! It would be fun to try to extend the map with bridges between islands, (perhaps adjoint functors might serve for this) and then talk of reefs that are almost above sea level, etc.
$endgroup$
– Tim Porter
Jun 8 at 14:02
add a comment
|
2
$begingroup$
Some of the 'islands' are very close to each other, so the analogy of a map has its limitations. Any compact metric (CM) space can be thought of as sitting in the Hilbert cube, and then it can be thought of as an intersection of polyhedral neighbourhoods so although CM spaces can be pathological from some points of view they are sort of infinitely near the polyhedral space island!!! It would be fun to try to extend the map with bridges between islands, (perhaps adjoint functors might serve for this) and then talk of reefs that are almost above sea level, etc.
$endgroup$
– Tim Porter
Jun 8 at 14:02
2
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Some of the 'islands' are very close to each other, so the analogy of a map has its limitations. Any compact metric (CM) space can be thought of as sitting in the Hilbert cube, and then it can be thought of as an intersection of polyhedral neighbourhoods so although CM spaces can be pathological from some points of view they are sort of infinitely near the polyhedral space island!!! It would be fun to try to extend the map with bridges between islands, (perhaps adjoint functors might serve for this) and then talk of reefs that are almost above sea level, etc.
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– Tim Porter
Jun 8 at 14:02
$begingroup$
Some of the 'islands' are very close to each other, so the analogy of a map has its limitations. Any compact metric (CM) space can be thought of as sitting in the Hilbert cube, and then it can be thought of as an intersection of polyhedral neighbourhoods so although CM spaces can be pathological from some points of view they are sort of infinitely near the polyhedral space island!!! It would be fun to try to extend the map with bridges between islands, (perhaps adjoint functors might serve for this) and then talk of reefs that are almost above sea level, etc.
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– Tim Porter
Jun 8 at 14:02
add a comment
|
$begingroup$
I'll go ahead and say that Polish spaces are an interesting and almost sui generis class. There is a rich literature of applications to and from descriptive set theory, with layers of "pathology" hierarchically organized along lines closely related to the arithmetical and analytic hierarchies. They are also widely used, as Nate mentioned, in abstract measure and probability theory.
I would say there are isthmuses connecting this class to the class of continua (mentioned by D.S. Lipham; consider for example the theory of pseudo-arcs) as compact connected metric spaces, and to some extent to locally compact Hausdorff spaces (e.g., a locally compact Hausdorff space is Polish iff it is second countable).
answered Jun 6 at 14:24
Todd Trimble♦Todd Trimble
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$begingroup$
I'll go ahead and say that Polish spaces are an interesting and almost sui generis class. There is a rich literature of applications to and from descriptive set theory, with layers of "pathology" hierarchically organized along lines closely related to the arithmetical and analytic hierarchies. They are also widely used, as Nate mentioned, in abstract measure and probability theory.
I would say there are isthmuses connecting this class to the class of continua (mentioned by D.S. Lipham; consider for example the theory of pseudo-arcs) as compact connected metric spaces, and to some extent to locally compact Hausdorff spaces (e.g., a locally compact Hausdorff space is Polish iff it is second countable).
answered Jun 6 at 14:24
Todd Trimble♦Todd Trimble
44.6k5 gold badges158 silver badges270 bronze badges
$endgroup$
add a comment
|
$begingroup$
I'll go ahead and say that Polish spaces are an interesting and almost sui generis class. There is a rich literature of applications to and from descriptive set theory, with layers of "pathology" hierarchically organized along lines closely related to the arithmetical and analytic hierarchies. They are also widely used, as Nate mentioned, in abstract measure and probability theory.
I would say there are isthmuses connecting this class to the class of continua (mentioned by D.S. Lipham; consider for example the theory of pseudo-arcs) as compact connected metric spaces, and to some extent to locally compact Hausdorff spaces (e.g., a locally compact Hausdorff space is Polish iff it is second countable).
answered Jun 6 at 14:24
Todd Trimble♦Todd Trimble
44.6k5 gold badges158 silver badges270 bronze badges
$endgroup$
I'll go ahead and say that Polish spaces are an interesting and almost sui generis class. There is a rich literature of applications to and from descriptive set theory, with layers of "pathology" hierarchically organized along lines closely related to the arithmetical and analytic hierarchies. They are also widely used, as Nate mentioned, in abstract measure and probability theory.
I would say there are isthmuses connecting this class to the class of continua (mentioned by D.S. Lipham; consider for example the theory of pseudo-arcs) as compact connected metric spaces, and to some extent to locally compact Hausdorff spaces (e.g., a locally compact Hausdorff space is Polish iff it is second countable).
answered Jun 6 at 14:24
Todd Trimble♦Todd Trimble
44.6k5 gold badges158 silver badges270 bronze badges
answered Jun 6 at 14:24
Todd Trimble♦Todd Trimble
44.6k5 gold badges158 silver badges270 bronze badges
answered Jun 6 at 14:24
Todd Trimble♦Todd Trimble
44.6k5 gold badges158 silver badges270 bronze badges
answered Jun 6 at 14:24
Todd Trimble♦Todd Trimble
44.6k5 gold badges158 silver badges270 bronze badges
44.6k5 gold badges158 silver badges270 bronze badges
add a comment
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$begingroup$
o-minimal structures, as generalisation of “tame” topology of semi-algebraic and semi-analytic sets, are quite interesting, in applications in particular. See e.g. the book
https://books.google.co.uk/books/about/Tame_Topology_and_O_minimal_Structures.html
answered Jun 6 at 16:05
Dima PasechnikDima Pasechnik
10.6k2 gold badges22 silver badges57 bronze badges
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$begingroup$
A great book which deserves to be well-known.
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– Todd Trimble♦
Jun 8 at 11:49
add a comment
|
$begingroup$
o-minimal structures, as generalisation of “tame” topology of semi-algebraic and semi-analytic sets, are quite interesting, in applications in particular. See e.g. the book
https://books.google.co.uk/books/about/Tame_Topology_and_O_minimal_Structures.html
answered Jun 6 at 16:05
Dima PasechnikDima Pasechnik
10.6k2 gold badges22 silver badges57 bronze badges
$endgroup$
$begingroup$
A great book which deserves to be well-known.
$endgroup$
– Todd Trimble♦
Jun 8 at 11:49
add a comment
|
$begingroup$
o-minimal structures, as generalisation of “tame” topology of semi-algebraic and semi-analytic sets, are quite interesting, in applications in particular. See e.g. the book
https://books.google.co.uk/books/about/Tame_Topology_and_O_minimal_Structures.html
answered Jun 6 at 16:05
Dima PasechnikDima Pasechnik
10.6k2 gold badges22 silver badges57 bronze badges
$endgroup$
o-minimal structures, as generalisation of “tame” topology of semi-algebraic and semi-analytic sets, are quite interesting, in applications in particular. See e.g. the book
https://books.google.co.uk/books/about/Tame_Topology_and_O_minimal_Structures.html
answered Jun 6 at 16:05
Dima PasechnikDima Pasechnik
10.6k2 gold badges22 silver badges57 bronze badges
answered Jun 6 at 16:05
Dima PasechnikDima Pasechnik
10.6k2 gold badges22 silver badges57 bronze badges
answered Jun 6 at 16:05
Dima PasechnikDima Pasechnik
10.6k2 gold badges22 silver badges57 bronze badges
answered Jun 6 at 16:05
Dima PasechnikDima Pasechnik
10.6k2 gold badges22 silver badges57 bronze badges
10.6k2 gold badges22 silver badges57 bronze badges
$begingroup$
A great book which deserves to be well-known.
$endgroup$
– Todd Trimble♦
Jun 8 at 11:49
add a comment
|
$begingroup$
A great book which deserves to be well-known.
$endgroup$
– Todd Trimble♦
Jun 8 at 11:49
$begingroup$
A great book which deserves to be well-known.
$endgroup$
– Todd Trimble♦
Jun 8 at 11:49
$begingroup$
A great book which deserves to be well-known.
$endgroup$
– Todd Trimble♦
Jun 8 at 11:49
add a comment
|
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8
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Stone spaces are Zariski spectra of Boolean rings.
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– Todd Trimble♦
Jun 6 at 5:00
9
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Polish spaces as another class?
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– Todd Trimble♦
Jun 6 at 5:17
7
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Another vote for Polish spaces from me. They're the typical "nice" space for measure theory, probability, and descriptive set theory. Elsewhere in analysis, "locally compact Hausdorff" is an important island.
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– Nate Eldredge
Jun 6 at 6:32
8
$begingroup$
The sea of pathology has various levels of depth: there is the deep sea of non-$T_0$ spaces, but there are deeper trenches: pretopological spaces and pseudotopological spaces. Of what unseen horrors in these depths may lurk few have lived to tell the tale. Also, let's not forget the great ocean of locales (connected by the strait of sober spaces) in which perhaps more islands of non-pathology can be found.
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– Gro-Tsen
Jun 6 at 12:00
5
$begingroup$
I find it difficult to see how $omega_1+1$ with its order topology is on an island (of Stone spaces) whereas inserting pieces of $mathbb R$ to make the long line pushes the space into the ocean.
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– Andreas Blass
Jun 6 at 17:01