Cleanup for Noetherian and Alexandrov properties

Author: prabau

properties/P000090.md

@@ -2,7 +2,7 @@
 uid: P000090
 name: Alexandrov
 refs:
-  - mr: 1711071
+  - zb: "0944.54018"
     name: Alexandroff spaces (F. G. Arenas)
   - wikipedia: Alexandrov_topology
     name: Alexandrov Topology on Wikipedia
@@ -16,20 +16,18 @@ An arbitrary intersection of open sets is open.
 This is equivalent to each of the following:
 
 * An arbitrary union of closed sets is closed.
-* Each $x \in X$ has a unique smallest neighborhood.
-* For each $A \subseteq X$, $A$ is open iff $A \cap Y$ is open in the subspace $Y$ for each finite $Y \subseteq X$.
+* Each $x \in X$ has a smallest neighborhood.
+* $X$ is *finitely generated*:
+For each $A \subseteq X$, $A$ is open iff $A \cap Y$ is open in the subspace $Y$ for each finite $Y \subseteq X$.
 * There is a preorder (i.e., a reflexive and transitive relation) $\preceq$ on $X$ such that the open sets are precisely the upward closed sets (i.e., sets $A\subseteq X$ such that $x\in A$ and $x\preceq y$ imply $y\in A$).
 
 See {{wikipedia:Alexandrov_topology}} for a more extensive list.
-
-See also Topospace's [Alexandrov space](http://topospaces.subwiki.org/wiki/Alexandrov_space) article.
-
-{{mr:1711071}} is available at <https://eudml.org/doc/120504>.
+See also {{zb:0944.54018}}.
 
 ----
 #### Meta-properties
 
 - This property is hereditary.
 - This property is preserved by arbitrary disjoint unions.
-- This property is preserved by box products.
 - This property is preserved by finite products.
+- This property is preserved by box products.

properties/P000208.md

@@ -14,18 +14,22 @@ refs:
     name: Answer to "Is the product of two Artinian topological spaces Artinian?"
 ---
 
-A space for which every subspace is compact.
+Every subset of $X$ is compact.
 
 Equivalently:
-- Every open subset is compact.
+- Every open set is compact.
 - The open sets satisfy the *ascending chain condition*: There is no infinite strictly increasing sequence $O_1 \subsetneq O_2 \subsetneq \cdots$ of open sets.
 - The closed sets satisfy the *descending chain condition*: There is no infinite strictly decreasing sequence $Y_1 \supsetneq Y_2 \supsetneq \cdots$ of closed sets.
+- Every nonempty collection of open sets has a maximal element.
+- Every nonempty collection of closed sets has a minimal element.
 
 Defined on page 5 of {{zb:1325.14001}}, and on page 248 of {{doi:10.1017/9781316543870}}. Also see the section on [Noetherian topological spaces](https://stacks.math.columbia.edu/tag/0050) from the Stacks project.
 
+Compare with {P226}.
+
 ----
 #### Meta-properties
 
 - This property is hereditary.
-- This property is preserved by finite products (see {{mathse:5118758}}).
+- This property is preserved by finite products (see Theorem 3 in {{mathse:5118758}}).
 - This property is preserved by finite disjoint unions.

theorems/T000826.md

@@ -6,4 +6,5 @@ then:
   P000090: true
 ---
 
-For any $x \in X$, the collection of open neighborhoods of $x$ must have a minimal element $U$. Since for any other open neighborhood $V$, $V \cap U$ is an open neighborhood as well, we must have $V \cap U \subseteq U$, thus $U = V \cap U\subseteq V$ from minimality of $U$, implying uniqueness of $U$. 
+For any $x \in X$, the collection of open neighborhoods of $x$ must have a minimal element $U$,
+which is necessarily contained in every other neighborhood of $x$.