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.
On Nov 5, 6:12 pm, taffer <djr...@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.
> On Nov 5, 6:12 pm, taffer <djr...@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?
> On 6 Nov, 00:38, Chip Eastham <hardm...@gmail.com> wrote:
> > On Nov 5, 6:12 pm, taffer <djr...@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?
> On 6 Nov, 13:24, taffer <djr...@bath.ac.uk> wrote:
> > On 6 Nov, 00:38, Chip Eastham <hardm...@gmail.com> wrote:
> > > On Nov 5, 6:12 pm, taffer <djr...@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?
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:
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.
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