 |
|
| Science Forum Index » Mathematics Forum » What is a generalization?... |
|
Page 1 of 1 |
|
| Author |
Message |
| taffer... |
 Posted: Thu Nov 05, 2009 1:12 pm |
|
|
|
Guest
|
In what follows, everything (posets, lattices, topological spaces) is
finite.
A family of sets gives rise naturally to a lattice. A lattice gives
rise naturally to a poset. A poset gives rise naturally to a
topological space. But a topological space is just a certain type of
set family, and thus set families generalize topological spaces.
All these generalizations are proper: not every lattice would induce a
set family, nor would every poset induce a lattice, nor would every
topological space induce a poset (only the T0 spaces).
Thus set families are a proper generalization of set families. |
|
|
| Back to top |
|
|
|
| Chip Eastham... |
| Posted: Thu Nov 05, 2009 2:38 pm |
|
|
|
Guest
|
On Nov 5, 6:12 pm, taffer <djr... at (no spam) bath.ac.uk> wrote:
[quote]In what follows, everything (posets, lattices, topological spaces) is
finite.
A family of sets gives rise naturally to a lattice. A lattice gives
rise naturally to a poset. A poset gives rise naturally to a
topological space. But a topological space is just a certain type of
set family, and thus set families generalize topological spaces.
All these generalizations are proper: not every lattice would induce a
set family, nor would every poset induce a lattice, nor would every
topological space induce a poset (only the T0 spaces).
Thus set families are a proper generalization of set families.
[/quote]
In your first step, you say a family of
sets "gives rise naturally to a lattice."
But only certain families of sets (such
as a power set) are examples of lattices.
So your argument breaks down at its first
link.
regards, chip |
|
|
| Back to top |
|
|
|
| taffer... |
| Posted: Fri Nov 06, 2009 3:24 am |
|
|
|
Guest
|
On 6 Nov, 00:38, Chip Eastham <hardm... at (no spam) gmail.com> wrote:
[quote]On Nov 5, 6:12 pm, taffer <djr... at (no spam) bath.ac.uk> wrote:
In what follows, everything (posets, lattices, topological spaces) is
finite.
A family of sets gives rise naturally to a lattice. A lattice gives
rise naturally to a poset. A poset gives rise naturally to a
topological space. But a topological space is just a certain type of
set family, and thus set families generalize topological spaces.
All these generalizations are proper: not every lattice would induce a
set family, nor would every poset induce a lattice, nor would every
topological space induce a poset (only the T0 spaces).
Thus set families are a proper generalization of set families.
In your first step, you say a family of
sets "gives rise naturally to a lattice."
But only certain families of sets (such
as a power set) are examples of lattices.
So your argument breaks down at its first
link.
regards, chip
[/quote]
How could I be so stupid? But nevermind, skip that step; every set
family gives rise to a poset. Every poset gives rise to a topological
space, and every topological space gives rise to a set family. So we
still have the set families generalize set families.
What I'm wondering is, what is the right way to think about this? |
|
|
| Back to top |
|
|
|
| taffer... |
| Posted: Fri Nov 06, 2009 3:35 am |
|
|
|
Guest
|
On 6 Nov, 13:24, taffer <djr... at (no spam) bath.ac.uk> wrote:
[quote]On 6 Nov, 00:38, Chip Eastham <hardm... at (no spam) gmail.com> wrote:
On Nov 5, 6:12 pm, taffer <djr... at (no spam) bath.ac.uk> wrote:
In what follows, everything (posets, lattices, topological spaces) is
finite.
A family of sets gives rise naturally to a lattice. A lattice gives
rise naturally to a poset. A poset gives rise naturally to a
topological space. But a topological space is just a certain type of
set family, and thus set families generalize topological spaces.
All these generalizations are proper: not every lattice would induce a
set family, nor would every poset induce a lattice, nor would every
topological space induce a poset (only the T0 spaces).
Thus set families are a proper generalization of set families.
In your first step, you say a family of
sets "gives rise naturally to a lattice."
But only certain families of sets (such
as a power set) are examples of lattices.
So your argument breaks down at its first
link.
regards, chip
How could I be so stupid? But nevermind, skip that step; every set
family gives rise to a poset. Every poset gives rise to a topological
space, and every topological space gives rise to a set family. So we
still have the set families generalize set families.
What I'm wondering is, what is the right way to think about this?
[/quote]
Actually, the reason we don't have an outright contradiction is that
the map from set families to posets is not injective, like the map
from posets to topological spaces is. When you go from set families to
posets, information is destroyed (hence the map is not injective).
Still, where does this leave the concept of "generalization" in this
context? Is it right to say that posets "generalize" set families? |
|
|
| Back to top |
|
|
|
| Chip Eastham... |
| Posted: Sat Nov 07, 2009 3:25 am |
|
|
|
Guest
|
On Nov 6, 8:35 am, taffer <djr... at (no spam) bath.ac.uk> wrote:
[quote]On 6 Nov, 13:24, taffer <djr... at (no spam) bath.ac.uk> wrote:
On 6 Nov, 00:38, Chip Eastham <hardm... at (no spam) gmail.com> wrote:
On Nov 5, 6:12 pm, taffer <djr... at (no spam) bath.ac.uk> wrote:
In what follows, everything (posets, lattices, topological spaces) is
finite.
A family of sets gives rise naturally to a lattice. A lattice gives
rise naturally to a poset. A poset gives rise naturally to a
topological space. But a topological space is just a certain type of
set family, and thus set families generalize topological spaces.
All these generalizations are proper: not every lattice would induce a
set family, nor would every poset induce a lattice, nor would every
topological space induce a poset (only the T0 spaces).
Thus set families are a proper generalization of set families.
In your first step, you say a family of
sets "gives rise naturally to a lattice."
But only certain families of sets (such
as a power set) are examples of lattices.
So your argument breaks down at its first
link.
regards, chip
How could I be so stupid? But nevermind, skip that step; every set
family gives rise to a poset. Every poset gives rise to a topological
space, and every topological space gives rise to a set family. So we
still have the set families generalize set families.
What I'm wondering is, what is the right way to think about this?
Actually, the reason we don't have an outright contradiction is that
the map from set families to posets is not injective, like the map
from posets to topological spaces is. When you go from set families to
posets, information is destroyed (hence the map is not injective).
Still, where does this leave the concept of "generalization" in this
context? Is it right to say that posets "generalize" set families?
[/quote]
If you want statements that are right or wrong,
you'll need to frame them with more rigor.
Perhaps a close approximation to what you are
trying to discuss can be framed in terms of
category theory. "Functors" are maps from one
category to another which preserve relations
among objects and arrows:
[Category theory -- Wikipedia]
http://en.wikipedia.org/wiki/Category_theory
Your revised discussion begins with a family
of sets being considered as a partial order
(by inclusion?), then proceeds to consider
a partial order giving rise to a topology
(the Alexandrov topology(?), where upper
sets are open), and finally considers the
topology as a family of (open?) sets.
Likely you mean to condition your discussion
on an assumption of finiteness throughout,
as in the original post.
It seems clear than this circle of ideas
does not necessarily bring us back to the
place we start. Suppose P is a partition
of a finite set, thus a collection of
disjoint finite sets. The partial order
consists (if my understanding of your
suggested construction is right) of just
incomparable points, and the topology is
a discrete topology. The open sets are
thus the power set of P, certainly not P
itself.
regards, chip |
|
|
| Back to top |
|
|
|
|
|
All times are GMT - 5 Hours
The time now is Fri Nov 27, 2009 2:18 am
|
|