update description

Author: yhx-12243

properties/P000227.md

@@ -1,18 +1,18 @@
 ---
 uid: P000227
-name: Has a discrete closed set of size $\mathfrak c$
+name: Has a discrete closed subset of size $\mathfrak c$
 ---
 
-$X$ has a discrete closed set of cardinality $\mathfrak c=2^{\aleph_0}$.
+$X$ has a discrete closed subset of cardinality $\mathfrak c=2^{\aleph_0}$.
 
-*Note*: Discrete closed sets are exactly the sets with no limit point in $X$.
+*Note*: Discrete closed subsets are exactly the subsets with no limit point in $X$.
 
 *Note*: This property implies $e(X)\ge\mathfrak c$,
-where the *extent* $e(X)$ is the supremum of the cardinality of discrete closed sets in $X$.
-But there are spaces with $e(X)=\mathfrak c$ and without discrete closed set of cardinality $\mathfrak c$,
-i.e., where the supremum is not attained.
+where the *extent* $e(X)$ is the supremum of the cardinality of discrete closed subsets in $X$.
+But there are spaces with $e(X)=\mathfrak c$ and without discrete closed subset of cardinality $\mathfrak c$,
+i.e., where the supremum is not attained, under certain set-theoretical assumptions.
 
-Compare with these properties, where $D$ denotes a discrete closed set in $X$:
+Compare with these properties, where $D$ denotes a discrete closed subset in $X$:
 - {P107} $(\exists D: |D|=1)$
 - {P21} $(\forall D: |D|<\aleph_0)$
 - {P198} $(\forall D: |D|\le\aleph_0)$