tweaks

properties/P000088.md

@@ -16,13 +16,13 @@ refs:
     name: Baireness of $C_k(X)$ for locally compact $X$ (Gruenhage & Ma)
 ---
 
-For every discrete family $(F_i)_{i \in I}$  of closed subsets of $X$, there exists a pairwise disjoint family of open sets $(U_i)_{i \in I}$, such that $F_i \subseteq U_i$ for all $i$.
+For every discrete family $(F_i)_{i \in I}$  of closed subsets of $X$, there exists a pairwise disjoint family of open sets $(U_i)_{i \in I}$ with $F_i \subseteq U_i$ for all $i$.
 
-Equivalently, for every discrete family $(F_i)_{i \in I}$  of closed subsets of $X$, there exists a discrete family of open sets $(U_i)_{i \in I}$, such that $F_i \subseteq U_i$ for all $i$.
+Equivalently, for every discrete family $(F_i)_{i \in I}$  of closed subsets of $X$, there exists a discrete family of open sets $(U_i)_{i \in I}$ with $F_i \subseteq U_i$ for all $i$.
 
 Terminology:
 - A *discrete family* is a family of subsets such that each point of $X$ has a neighborhood meeting at most one of the subsets.
-- In the first definition, the family of open sets $(U_i)_i$ is sometimes called a *disjoint open expansion* of the family $(F_i)_i$. (REFERENCE ?)
+- In the first definition, the family of open sets $(U_i)_i$ is sometimes called a *disjoint open expansion* of the family $(F_i)_i$.
 - In the second definition, the family of open sets $(U_i)_i$ is sometimes called a *discrete open expansion* of the family $(F_i)_i$. (See for example {{doi:10.1016/S0166-8641(96)00163-0}}.)
 
 The equivalence between the two definitions is Theorem 5.1.17 of {{zb:0684.54001}}.

spaces/S000082/properties/P000021.md

@@ -8,4 +8,5 @@ refs:
 ---
 
 Every non-empty set has a limit point, since every non-empty subset of
-{S42} has a limit point (e.g. any lesser point).
+{S42} has a limit point
+(e.g. any point smaller than a point in the set).

theorems/T000657.md → theorems/T000661.md

@@ -1,5 +1,5 @@
 ---
-uid: T000657
+uid: T000661
 if:
   and:
   - P000197: true

theorems/T000658.md → theorems/T000662.md

@@ -1,5 +1,5 @@
 ---
-uid: T000658
+uid: T000662
 if:
   and:
   - P000198: true