What are the 'spaces' whose structure is defined by a collection of subsets Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Why are convex sets assumed / defined to be subsets of vector spaces (and not of more general spaces)?Mathematical structuresIs there a formal notion of equivalence between structures with potentially different signatures?A pedantic question about defining new structures in a path-independent way.Why are topological spaces interesting to study?Why are empty measurable spaces and empty topological spaces not desirable?Neighborhood Topology and Open Set Topology: their Equivalence and ComparisonWhy does topology rarely come up outside of topology?Algebraic structure of the extended real line $overlineBbb R$.Are topoi like topological spaces or like set theory?
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What are the 'spaces' whose structure is defined by a collection of subsets
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Why are convex sets assumed / defined to be subsets of vector spaces (and not of more general spaces)?Mathematical structuresIs there a formal notion of equivalence between structures with potentially different signatures?A pedantic question about defining new structures in a path-independent way.Why are topological spaces interesting to study?Why are empty measurable spaces and empty topological spaces not desirable?Neighborhood Topology and Open Set Topology: their Equivalence and ComparisonWhy does topology rarely come up outside of topology?Algebraic structure of the extended real line $overlineBbb R$.Are topoi like topological spaces or like set theory?
$begingroup$
I am trying to recount which 'space' structures are defined by a collection of subsets. Given a set $X$ the only two structures I know are:
(1) $mathcalTsubseteq mathcalP(X)$ is a topology on $X$, and $(X,mathcalT)$ is a topological space.
(2) $mathcalFsubseteq mathcalP(X)$ is a $sigma$-algebra on $X$, and $(X,mathcalF)$ is a measurable space.
Are there any more such famous structures given by a collection of subsets?
soft-question recreational-mathematics
asked Mar 26 at 8:05
Keen-ameteurKeen-ameteur
1,550516
$endgroup$
add a comment |
$begingroup$
I am trying to recount which 'space' structures are defined by a collection of subsets. Given a set $X$ the only two structures I know are:
(1) $mathcalTsubseteq mathcalP(X)$ is a topology on $X$, and $(X,mathcalT)$ is a topological space.
(2) $mathcalFsubseteq mathcalP(X)$ is a $sigma$-algebra on $X$, and $(X,mathcalF)$ is a measurable space.
Are there any more such famous structures given by a collection of subsets?
soft-question recreational-mathematics
asked Mar 26 at 8:05
Keen-ameteurKeen-ameteur
1,550516
$endgroup$
add a comment |
$begingroup$
I am trying to recount which 'space' structures are defined by a collection of subsets. Given a set $X$ the only two structures I know are:
(1) $mathcalTsubseteq mathcalP(X)$ is a topology on $X$, and $(X,mathcalT)$ is a topological space.
(2) $mathcalFsubseteq mathcalP(X)$ is a $sigma$-algebra on $X$, and $(X,mathcalF)$ is a measurable space.
Are there any more such famous structures given by a collection of subsets?
soft-question recreational-mathematics
asked Mar 26 at 8:05
Keen-ameteurKeen-ameteur
1,550516
$endgroup$
I am trying to recount which 'space' structures are defined by a collection of subsets. Given a set $X$ the only two structures I know are:
(1) $mathcalTsubseteq mathcalP(X)$ is a topology on $X$, and $(X,mathcalT)$ is a topological space.
(2) $mathcalFsubseteq mathcalP(X)$ is a $sigma$-algebra on $X$, and $(X,mathcalF)$ is a measurable space.
Are there any more such famous structures given by a collection of subsets?
soft-question recreational-mathematics
soft-question recreational-mathematics
asked Mar 26 at 8:05
Keen-ameteurKeen-ameteur
1,550516
asked Mar 26 at 8:05
Keen-ameteurKeen-ameteur
1,550516
asked Mar 26 at 8:05
Keen-ameteurKeen-ameteur
1,550516
asked Mar 26 at 8:05
Keen-ameteurKeen-ameteur
1,550516
asked Mar 26 at 8:05
Keen-ameteurKeen-ameteur
1,550516
1,550516
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
In measure theory there are often technical such structures (semi-algebra/algebra on a set $X$, $pi$-system, $lambda$-system) that are used in the development of the theory, but are rarely studied independently, so I'll ignore them...
A bornological space ("bounded sets") is another one occurring in analysis.
A more geometric one is abstract convexity structures (where we have an axiomatisation of convex sets, but there are also metric convexities, order convexities etc. see the standard reference book by van de Vel for much more info. These are closed under arbitary intersections, contain $emptyset$ and $X$, and are closed under all up-directed unions, to summarise the axioms. It's very much in the spirit of topology (separation axioms etc.), IMHO it's a nice theory that deserves more attention.
Stretching it a bit, (undirected) graphs are sets with special "doubleton sets" (the edges), but these have no special axioms. One can generalise to multigraphs and incidence structures and matroids (the "independent" subsets). The last one are a proper example in your sense, I think.
Maybe others can come up with more than bornological spaces, convexity structures and matroids.
answered Mar 26 at 9:44
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
Thank for your extensive answer. Do these objects also give rise to the term of generated structure, as in minimal such sturctures containing a given collection? For example generated topology, or generated $sigma$-algebra.
$endgroup$
– Keen-ameteur
Mar 26 at 10:19
2
$begingroup$
@Keen-ameteur often they do, I'm not too familiar with most of them. Such induced structures are very common in categorial topology, so in closure spaces, uniform spaces, convergence spaces, nearness spaces, proximity spaces etc etc. They almost all have such notions like initial and final structures induced by maps.
$endgroup$
– Henno Brandsma
Mar 26 at 10:23
add a comment |
$begingroup$
The most important "space-like" structure defined on subsets in the realm of combinatorial commutative algebra is the (abstract) simplicial complex. These correspond in a natural way to square-free monomial ideals. This correspondence unleashes a bevy of tools from commutative algebra to study the combinatorics of simplicial complexes and vice versa.
As hinted at by @Henno, matroids form a particular family of simplicial complexes (via their independent sets). Greedoids are a generalization of matroids that are "spaces" defined on power sets of a given (finite) set that behave well with respect to the greedy algorithm. One class of greedoids are the antimatroids which are complementary combinatorial objects to convex geometries.
What all of these examples have in common is that they are combinatorial abstractions of some classically studied structure usually embedded in some (typically Euclidean) geometry. Simplicial complexes are abstractions of triangulated topological spaces; matroids capture all of the (linear) dependencies between vectors in a given finite subset of a vector space; convex geometries generalize shellings of polytopes.
answered Mar 28 at 20:34
Aaron DallAaron Dall
586416
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
In measure theory there are often technical such structures (semi-algebra/algebra on a set $X$, $pi$-system, $lambda$-system) that are used in the development of the theory, but are rarely studied independently, so I'll ignore them...
A bornological space ("bounded sets") is another one occurring in analysis.
A more geometric one is abstract convexity structures (where we have an axiomatisation of convex sets, but there are also metric convexities, order convexities etc. see the standard reference book by van de Vel for much more info. These are closed under arbitary intersections, contain $emptyset$ and $X$, and are closed under all up-directed unions, to summarise the axioms. It's very much in the spirit of topology (separation axioms etc.), IMHO it's a nice theory that deserves more attention.
Stretching it a bit, (undirected) graphs are sets with special "doubleton sets" (the edges), but these have no special axioms. One can generalise to multigraphs and incidence structures and matroids (the "independent" subsets). The last one are a proper example in your sense, I think.
Maybe others can come up with more than bornological spaces, convexity structures and matroids.
answered Mar 26 at 9:44
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
Thank for your extensive answer. Do these objects also give rise to the term of generated structure, as in minimal such sturctures containing a given collection? For example generated topology, or generated $sigma$-algebra.
$endgroup$
– Keen-ameteur
Mar 26 at 10:19
2
$begingroup$
@Keen-ameteur often they do, I'm not too familiar with most of them. Such induced structures are very common in categorial topology, so in closure spaces, uniform spaces, convergence spaces, nearness spaces, proximity spaces etc etc. They almost all have such notions like initial and final structures induced by maps.
$endgroup$
– Henno Brandsma
Mar 26 at 10:23
add a comment |
$begingroup$
In measure theory there are often technical such structures (semi-algebra/algebra on a set $X$, $pi$-system, $lambda$-system) that are used in the development of the theory, but are rarely studied independently, so I'll ignore them...
A bornological space ("bounded sets") is another one occurring in analysis.
A more geometric one is abstract convexity structures (where we have an axiomatisation of convex sets, but there are also metric convexities, order convexities etc. see the standard reference book by van de Vel for much more info. These are closed under arbitary intersections, contain $emptyset$ and $X$, and are closed under all up-directed unions, to summarise the axioms. It's very much in the spirit of topology (separation axioms etc.), IMHO it's a nice theory that deserves more attention.
Stretching it a bit, (undirected) graphs are sets with special "doubleton sets" (the edges), but these have no special axioms. One can generalise to multigraphs and incidence structures and matroids (the "independent" subsets). The last one are a proper example in your sense, I think.
Maybe others can come up with more than bornological spaces, convexity structures and matroids.
answered Mar 26 at 9:44
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
Thank for your extensive answer. Do these objects also give rise to the term of generated structure, as in minimal such sturctures containing a given collection? For example generated topology, or generated $sigma$-algebra.
$endgroup$
– Keen-ameteur
Mar 26 at 10:19
2
$begingroup$
@Keen-ameteur often they do, I'm not too familiar with most of them. Such induced structures are very common in categorial topology, so in closure spaces, uniform spaces, convergence spaces, nearness spaces, proximity spaces etc etc. They almost all have such notions like initial and final structures induced by maps.
$endgroup$
– Henno Brandsma
Mar 26 at 10:23
add a comment |
$begingroup$
In measure theory there are often technical such structures (semi-algebra/algebra on a set $X$, $pi$-system, $lambda$-system) that are used in the development of the theory, but are rarely studied independently, so I'll ignore them...
A bornological space ("bounded sets") is another one occurring in analysis.
A more geometric one is abstract convexity structures (where we have an axiomatisation of convex sets, but there are also metric convexities, order convexities etc. see the standard reference book by van de Vel for much more info. These are closed under arbitary intersections, contain $emptyset$ and $X$, and are closed under all up-directed unions, to summarise the axioms. It's very much in the spirit of topology (separation axioms etc.), IMHO it's a nice theory that deserves more attention.
Stretching it a bit, (undirected) graphs are sets with special "doubleton sets" (the edges), but these have no special axioms. One can generalise to multigraphs and incidence structures and matroids (the "independent" subsets). The last one are a proper example in your sense, I think.
Maybe others can come up with more than bornological spaces, convexity structures and matroids.
answered Mar 26 at 9:44
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
In measure theory there are often technical such structures (semi-algebra/algebra on a set $X$, $pi$-system, $lambda$-system) that are used in the development of the theory, but are rarely studied independently, so I'll ignore them...
A bornological space ("bounded sets") is another one occurring in analysis.
A more geometric one is abstract convexity structures (where we have an axiomatisation of convex sets, but there are also metric convexities, order convexities etc. see the standard reference book by van de Vel for much more info. These are closed under arbitary intersections, contain $emptyset$ and $X$, and are closed under all up-directed unions, to summarise the axioms. It's very much in the spirit of topology (separation axioms etc.), IMHO it's a nice theory that deserves more attention.
Stretching it a bit, (undirected) graphs are sets with special "doubleton sets" (the edges), but these have no special axioms. One can generalise to multigraphs and incidence structures and matroids (the "independent" subsets). The last one are a proper example in your sense, I think.
Maybe others can come up with more than bornological spaces, convexity structures and matroids.
answered Mar 26 at 9:44
Henno BrandsmaHenno Brandsma
116k349127
answered Mar 26 at 9:44
Henno BrandsmaHenno Brandsma
116k349127
answered Mar 26 at 9:44
Henno BrandsmaHenno Brandsma
116k349127
answered Mar 26 at 9:44
Henno BrandsmaHenno Brandsma
116k349127
116k349127
$begingroup$
Thank for your extensive answer. Do these objects also give rise to the term of generated structure, as in minimal such sturctures containing a given collection? For example generated topology, or generated $sigma$-algebra.
$endgroup$
– Keen-ameteur
Mar 26 at 10:19
2
$begingroup$
@Keen-ameteur often they do, I'm not too familiar with most of them. Such induced structures are very common in categorial topology, so in closure spaces, uniform spaces, convergence spaces, nearness spaces, proximity spaces etc etc. They almost all have such notions like initial and final structures induced by maps.
$endgroup$
– Henno Brandsma
Mar 26 at 10:23
add a comment |
$begingroup$
Thank for your extensive answer. Do these objects also give rise to the term of generated structure, as in minimal such sturctures containing a given collection? For example generated topology, or generated $sigma$-algebra.
$endgroup$
– Keen-ameteur
Mar 26 at 10:19
2
$begingroup$
@Keen-ameteur often they do, I'm not too familiar with most of them. Such induced structures are very common in categorial topology, so in closure spaces, uniform spaces, convergence spaces, nearness spaces, proximity spaces etc etc. They almost all have such notions like initial and final structures induced by maps.
$endgroup$
– Henno Brandsma
Mar 26 at 10:23
$begingroup$
Thank for your extensive answer. Do these objects also give rise to the term of generated structure, as in minimal such sturctures containing a given collection? For example generated topology, or generated $sigma$-algebra.
$endgroup$
– Keen-ameteur
Mar 26 at 10:19
$begingroup$
Thank for your extensive answer. Do these objects also give rise to the term of generated structure, as in minimal such sturctures containing a given collection? For example generated topology, or generated $sigma$-algebra.
$endgroup$
– Keen-ameteur
Mar 26 at 10:19
2
2
$begingroup$
@Keen-ameteur often they do, I'm not too familiar with most of them. Such induced structures are very common in categorial topology, so in closure spaces, uniform spaces, convergence spaces, nearness spaces, proximity spaces etc etc. They almost all have such notions like initial and final structures induced by maps.
$endgroup$
– Henno Brandsma
Mar 26 at 10:23
$begingroup$
@Keen-ameteur often they do, I'm not too familiar with most of them. Such induced structures are very common in categorial topology, so in closure spaces, uniform spaces, convergence spaces, nearness spaces, proximity spaces etc etc. They almost all have such notions like initial and final structures induced by maps.
$endgroup$
– Henno Brandsma
Mar 26 at 10:23
add a comment |
$begingroup$
The most important "space-like" structure defined on subsets in the realm of combinatorial commutative algebra is the (abstract) simplicial complex. These correspond in a natural way to square-free monomial ideals. This correspondence unleashes a bevy of tools from commutative algebra to study the combinatorics of simplicial complexes and vice versa.
As hinted at by @Henno, matroids form a particular family of simplicial complexes (via their independent sets). Greedoids are a generalization of matroids that are "spaces" defined on power sets of a given (finite) set that behave well with respect to the greedy algorithm. One class of greedoids are the antimatroids which are complementary combinatorial objects to convex geometries.
What all of these examples have in common is that they are combinatorial abstractions of some classically studied structure usually embedded in some (typically Euclidean) geometry. Simplicial complexes are abstractions of triangulated topological spaces; matroids capture all of the (linear) dependencies between vectors in a given finite subset of a vector space; convex geometries generalize shellings of polytopes.
answered Mar 28 at 20:34
Aaron DallAaron Dall
586416
$endgroup$
add a comment |
$begingroup$
The most important "space-like" structure defined on subsets in the realm of combinatorial commutative algebra is the (abstract) simplicial complex. These correspond in a natural way to square-free monomial ideals. This correspondence unleashes a bevy of tools from commutative algebra to study the combinatorics of simplicial complexes and vice versa.
As hinted at by @Henno, matroids form a particular family of simplicial complexes (via their independent sets). Greedoids are a generalization of matroids that are "spaces" defined on power sets of a given (finite) set that behave well with respect to the greedy algorithm. One class of greedoids are the antimatroids which are complementary combinatorial objects to convex geometries.
What all of these examples have in common is that they are combinatorial abstractions of some classically studied structure usually embedded in some (typically Euclidean) geometry. Simplicial complexes are abstractions of triangulated topological spaces; matroids capture all of the (linear) dependencies between vectors in a given finite subset of a vector space; convex geometries generalize shellings of polytopes.
answered Mar 28 at 20:34
Aaron DallAaron Dall
586416
$endgroup$
add a comment |
$begingroup$
The most important "space-like" structure defined on subsets in the realm of combinatorial commutative algebra is the (abstract) simplicial complex. These correspond in a natural way to square-free monomial ideals. This correspondence unleashes a bevy of tools from commutative algebra to study the combinatorics of simplicial complexes and vice versa.
As hinted at by @Henno, matroids form a particular family of simplicial complexes (via their independent sets). Greedoids are a generalization of matroids that are "spaces" defined on power sets of a given (finite) set that behave well with respect to the greedy algorithm. One class of greedoids are the antimatroids which are complementary combinatorial objects to convex geometries.
What all of these examples have in common is that they are combinatorial abstractions of some classically studied structure usually embedded in some (typically Euclidean) geometry. Simplicial complexes are abstractions of triangulated topological spaces; matroids capture all of the (linear) dependencies between vectors in a given finite subset of a vector space; convex geometries generalize shellings of polytopes.
answered Mar 28 at 20:34
Aaron DallAaron Dall
586416
$endgroup$
The most important "space-like" structure defined on subsets in the realm of combinatorial commutative algebra is the (abstract) simplicial complex. These correspond in a natural way to square-free monomial ideals. This correspondence unleashes a bevy of tools from commutative algebra to study the combinatorics of simplicial complexes and vice versa.
As hinted at by @Henno, matroids form a particular family of simplicial complexes (via their independent sets). Greedoids are a generalization of matroids that are "spaces" defined on power sets of a given (finite) set that behave well with respect to the greedy algorithm. One class of greedoids are the antimatroids which are complementary combinatorial objects to convex geometries.
What all of these examples have in common is that they are combinatorial abstractions of some classically studied structure usually embedded in some (typically Euclidean) geometry. Simplicial complexes are abstractions of triangulated topological spaces; matroids capture all of the (linear) dependencies between vectors in a given finite subset of a vector space; convex geometries generalize shellings of polytopes.
answered Mar 28 at 20:34
Aaron DallAaron Dall
586416
answered Mar 28 at 20:34
Aaron DallAaron Dall
586416
answered Mar 28 at 20:34
Aaron DallAaron Dall
586416
answered Mar 28 at 20:34
Aaron DallAaron Dall
586416
586416
add a comment |
add a comment |
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