prabau
commented
20 hours ago Let me try to explore the concepts related to the specialization preorder on a space $X$.
Specialization/Generalization relation
Let $x,y\in X$. If $x\in cl\{y\}$ (equivalently, $cl\{x\}\subseteq cl\{y\}$), one says:
- $x$ is a specialization of $y$
- $y$ is a generalization of $x$
The notation for this is always the "squiggly arrow": $y\rightsquigarrow x$ (it goes from more generalized point to more specialized point).
The relation "is a specialization of" is a preorder. The relation "is a generalization of" is the reverse preorder.
Now, as explained in the wikipedia page, there are two conventions when defining the "specialization preorder"; it is meant either as the "is a specialization of" preorder, or the "is a generalization of" preorder.
For definiteness, let's pick $x\le y$ to mean $x$ is a specialization of $y$ (i.e., $y\rightsquigarrow x$). As a particular example, given a preorder $(X,\le)$, one can form the Alexandrov topology on $X$ with the open sets being the upper sets. The more specialized points will be further down and the less specialized points will be further up.
Maximum, maximal, etc
Given any preorder $(X,\le)$, one says that two elements $x$ and $y$ are equivalent (notation: $x\sim y$) if $x\le y$ and $y\le x$. (For the specialization preorder, that would correspond to $cl\{x\}=cl\{y\}$, that is, $x$ and $y$ are topologically indistinguishable.)
One also defines:
- $x$ is a maximum element if $y\le x$ for all $y\in X$.
- $x$ is a maximal element if any element $y\ge x$ is necessarily equivalent to $x$.
- $x$ is a minimum element if $y\ge x$ for all $y\in X$.
- $x$ is a minimal element if any element $y\le x$ is necessarily equivalent to $x$.
Applying this to the specialization preorder
Let $(X,\le)$ be the specialization preorder (so $x\le y$ iff $y\rightsquigarrow x$).
(1) Some equivalent formulations for " $x$ is maximum element for $\le$":
- $x$ is a generic point
- $x$ is a generalization of every other point
- Every point is a specialization of $x$
- $x$ is more generalized than any other point (where "more generalized" means $\ge$)
- $x$ is less specialized than any other point (where "less specialized" means $\ge$)
- $x$ is a least specialized point (? ambiguous: could be interpreted as (2) below)
- $x$ is a most generalized point (? ambiguous, see (2))
(2) Some equivalent formulations for " $x$ is maximal element for $\le$":
- There is no element strictly more generalized than $x$
- Any generalization of $x$ is topologically indistinguishable from $x$
- $x$ is a most generalized point (? ambiguous, see (1))
(3) Some equivalent formulations for " $x$ is minimum element for $\le$":
- $x$ is a ???
- $x$ is a specialization of every other point
- Every point is a generalization of $x$
- $x$ is more specialized than any other point (where "more specialized" means $\le$)
- $x$ is less generalized than any other point (where "less generalized" means $\le$)
- $x$ is a least generalized point (? ambiguous: could be interpreted as (4) below)
- $x$ is a most specialized point (? ambiguous, see (4))
(4) Some equivalent formulations for " $x$ is minimal element for $\le$":
- There is no element strictly more specialized than $x$
- Any specialization of $x$ is topologically indistinguishable from $x$
- $x$ is a most specialized point (? ambiguous, see (3))
In light of all this, since "$x$ is a most specialized point" already feels ambiguous between (3) and (4), it looks that "$x$ is a specialized point" has an even less clear meaning. Every point is a specialized point of some point, maybe of itself, maybe of elements topologically indistinguishable from it, maybe of still other points, even if the point does not satisfy (3).
GeoffreySangston
StevenClontz
danflapjax
See discussion at #856.
I feel like the dual property "Has a point with a unique neighborhood" of "Has a generic point" needs a more obvious alias representing that duality. So while "Has a specialized point" is not in the literature, I wonder if it's appropriate to add it to $\pi$-Base as an alias (not our canonical name)
In #856 there was discussion of "Has a most specialized point". But since we don't say "Has a most generic point", maybe "Has a specialized point" is fine?