Consolidate use of Ishikawa's result (#1221)

properties/P000033.md

@@ -4,8 +4,18 @@ name: Countably metacompact
 refs:
   - doi: 10.1007/978-1-4612-6290-9
     name: Counterexamples in Topology
+  - zb: "0066.41001"
+    name: On Countably Paracompact Spaces (F. Ishikawa)
 ---
 
 Every countable open cover of the space has a point-finite open refinement.
 
+Equivalently (*Ishikawa's characterization*),
+for any nested sequence of closed sets $F_0 \supseteq F_1 \supseteq \cdots$ with empty intersection,
+there exist open sets $U_n \supseteq F_n$ for each $n$ such that the $U_n$ have empty intersection.
+
+(One can choose the $U_n$ to be nested if desired.)
+
+For a proof of the equivalence, see the Corollary in {{zb:0066.41001}}.
+
 Defined on page 23 of {{doi:10.1007/978-1-4612-6290-9}}.

spaces/S000061/README.md

@@ -8,7 +8,7 @@ refs:
   - doi: 10.1007/978-1-4612-6290-9 
     name: Counterexamples in Topology
 ---
-Let $D = \mathbb{R} \setminus \mathbb{Q}$. The pointed irrational extension topology on $\mathbb{R}$ is generated by the standard Euclidean open sets along with sets of the form $\{x\} \cup (D \cap U)$ where $U \subset \mathbb{R}$ is open and $x \in U$.
+Let $D = \mathbb{R} \setminus \mathbb{Q}$. The pointed irrational extension topology on $\mathbb{R}$ is generated by the sets of the form $\{x\} \cup (D \cap U)$ where $U\subset\mathbb R$ is open in the Euclidean topology and $x \in U$.
 
 Defined as counterexample #69 ("Pointed Irrational Extension of $\mathbb{R}$")
 in {{doi:10.1007/978-1-4612-6290-9}}.

spaces/S000061/properties/P000033.md

@@ -7,18 +7,20 @@ refs:
   name: Counterexamples in Topology
 ---
 
-Asserted in the General Reference Chart for space #69 in
-{{doi:10.1007/978-1-4612-6290-9_6}}.
+We will use Ishikawa's characterization of {P33}.
 
-Here is a direct proof of this fact:
+Let $q_0, q_1, \dots$ be an enumeration of the rational numbers $\mathbb Q$.
+Define $F_n := \{ q_i \mid i > n \}$.
+Each $F_n$ is closed in $X$ and the $F_n$ are nested with empty intersection.
+Moreover, $F_n$ is dense in $\mathbb R$ with respect to the usual Euclidean topology.
 
-As stated in the Corollary in {{zb:0066.41001}}, a necessary and sufficient condition for $X$ to be {P33} is that, for any nested sequence of closed sets $F_0 \supseteq F_1 \supseteq \cdots$ such that $\bigcap_{n \in \omega} F_n = \varnothing$, there exist open sets $U_0 \supseteq U_1 \supseteq \cdots$ such that $U_i \supseteq F_i$ for each $i$ and $\bigcap_{n \in \omega} U_n = \varnothing$.
+Note that the subspace topology on $D=\mathbb R\setminus\mathbb Q$ coincides with the Euclidean topology induced from $\mathbb R$.
+So the subspace $D$ is exactly {S28}.
 
-Let $q_0, q_1, \dots$ be a permutation of rational numbers $\mathbb Q$. Define $F_n := \left\{ q_i \mid i > n \right\}$, then $\bigcap_{n \in \omega} F_n = \varnothing$.
-Additionally, $F_n$ is closed because the subspace topology of $\mathbb Q$ is {P52}.
-Moreover, $F_n$ is dense in $\mathbb R$ with respect to its usual topology.
+Now, suppose $U_n$ is an open set containing $F_n$.
+The intersection $U_n \cap D$ is open in $D$ and it is easy to check it is dense in $D$.
 
-Now, suppose $U_n$ is an open set that contains $F_n$.
-Then, the intersection $U_n \cap (\mathbb R \setminus \mathbb Q)$ is a dense open subset of $\mathbb R \setminus \mathbb Q$ when considered with the subspace topology, which coincides with {S28}.
 
-Since {S28|P64}, $\bigcap_{n \in \omega} U_n \cap (\mathbb R \setminus \mathbb Q) \neq \varnothing$. This implies that {S61} is not {P33}.
+Since {S28|P64}, $\bigcap_{n \in \omega} (U_n \cap D)\ne\varnothing$,
+and hence $\bigcap_{n \in \omega}U_n\ne\varnothing$.
+This proves that the space is not {P33}.

theorems/T000388.md

@@ -4,17 +4,16 @@ if:
   P000132: true
 then:
   P000033: true
-refs:
-  - zb: "0066.41001"
-    name: On Countably Paracompact Spaces (F. Ishikawa)
 ---
 
-As stated in the Corollary in {{zb:0066.41001}}, a necessary and sufficient condition for $X$ to be {P33} is that, for any nested sequence of closed sets $F_0 \supseteq F_1 \supseteq \cdots$ such that $\bigcap_{n \in \omega} F_n = \varnothing$, there exist open sets $U_0 \supseteq U_1 \supseteq \cdots$ such that $U_i \supseteq F_i$ for each $i$ and $\bigcap_{n \in \omega} U_n = \varnothing$.
+We will use Ishikawa's characterization of {P33}.
 
-Now let $X$ be {P132} and $F_0 \supseteq F_1 \supseteq \cdots$ be an arbitrary nested sequence of closed sets.
-Since $F_n$ is a $G_\delta$ set, $F_n = \bigcap_{m \in \omega} U_{n, m}$ for open sets $U_{n, m}$.
+Suppose $X$ is {P132}.
+Let $F_0 \supseteq F_1 \supseteq \cdots$ be a nested sequence of closed sets with empty intersection.
+Since each $F_n$ is a $G_\delta$ set, $F_n = \bigcap_{m \in \omega} U_{n, m}$ for some open sets $U_{n, m}$.
 
-Define $U_k := \bigcap_{n \leq k, m \leq k} U_{n, m}$. Clearly $\left\{ U_n \right\}_{n \in \omega}$ is a decreasing sequence of open sets and $F_k = \bigcap_{n \leq k} F_n = \bigcap_{n \leq k, m \in \omega} U_{n, m} \subseteq U_k$.
+Define $U_k := \bigcap_{n \leq k, m \leq k} U_{n, m}$.
+Clearly each $U_k$ is an open set and $F_k = \bigcap_{n \leq k} F_n = \bigcap_{n \leq k, m \in \omega} U_{n, m} \subseteq U_k$.
 Finally, $\bigcap_{k \in \omega} U_k = \bigcap_{n \in \omega, m \in \omega} U_{n, m} = \bigcap_{m \in \omega} F_n = \varnothing$.
 
 Therefore, $X$ is {P33}.