---- > [!definition] Definition. ([[the specialization preorder on a topological space]]) > The **specialization (or canonical) preorder** on a [[topological space]] $(X, \tau)$ is defined by declaring $x \leq y \iff x \in \overline{\{ y \}},$ > where $\overline{\{ y \}}$ denotes the [[closure]] of $\{ y \}$. We say $x$ is a **specialization** of $y$.[^1] > > If $X$ satisfies the [[topologically distinguishable|T0 Axiom]], this [[poset|preorder]] is in fact a [[poset|partial order]], and is called the **specialization order**. ^definition > [!definition] (The specialization functor $\mathscr{W}$) > Any [[continuous|continuous map]] $f:X \to Y$ is [[monotonic map|monotone]] with respect to the specialization preorder, giving rise to a "specialization [[covariant functor|functor]]" $\mathscr{W}:\mathsf{Top} \to \mathsf{PrePos}$. Restricting $\mathscr{W}$ to the [[subcategory|(full sub)]][[category]] of [[Alexandrov topology|Alexandrov topological spaces]] [[the category of preposets is equivalent to that of Alexandrov spaces|witnesses]] a [[category equivalence]] $\mathsf{AlexTop} \xrightarrow{\sim} \mathsf{PrePos}$. ^definition > [!equivalence] > Unpacking the definition, one has $x \leq y$ if and only if $x$ belongs to every [[closed set]] containing $y$. Equivalently, $x \leq y$ if and only if $y$ belongs to every *open* set containing (that is, every [[neighborhood]] of) $x$. > > This is the sense in which $y$ is 'more general' than $x$: it is contained in more open sets. ^equivalence > [!equivalence] Equivalence for Alexandrov Topological spaces. > If $X$ is [[Alexandrov topology|Alexandrov]], then for all $x, y \in X$, $x \leq y \iff U_{x} \supset U_{y}, $ > where $U_{x}$ and $U_{y}$ denote the minimal open [[neighborhood|neighborhoods]] of $x$ and $y$ respectively. This 'order reversal' accounts for why the restriction maps for a cellular sheaf as defined e.g. in Curry 'go upward'. > > > [!proof]- > > This is easy to see from the first equivalence. > > > > $\implies$. If $x \leq y$, then $y$ belongs to every open set containing $x$, hence $y \in U_{x}$. Since $U_{y}$ equals the intersection of all open neighborhoods of $y$, and $U_{x}$ is one such, $U_{y} \subset U_{x}$. > > > > $\impliedby$. If $U_{x} \supset U_{y}$, and $U$ is an open set containing $x$, then $U \supset U_{x} \supset U_{y} \ni y$. > > > > [!justification] > This is indeed a [[poset|preorder]]: certainly $x \in \overline{\{ x \}}$ for all $x \in X$, and if $x \in \overline{\{ y \}}$ and $y \in \overline{\{ z \}}$ then $x \in \overline{\{ x \}} \subset \overline{\{ y \}} \subset \overline{ \overline{\{ z \}}}=\overline{\{ z \}}$. > If $X$ satisfies the [[topologically distinguishable|T0 Axiom]], we have both $x \leq y$ and $y \leq x$, then in fact $x=y$. Indeed, $x \leq y$ implies (cf. the equivalence) that $y$ belongs to every [[neighborhood]] of $x$. Meanwhile, $y \leq x$ implies that $x$ belongs to every [[neighborhood]] of $y$. Two points in a $T_{0}$ space having the same [[neighborhood|neighborhoods]] are equal. - [ ] check monotonicity of continuous maps ---- #### [^1]: Some conventions (e.g. Hartshorne's) instead say $y$ is a specialization of $x$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```