Updates for alpha_i properties

Author: prabau

properties/P000210.md

@@ -1,20 +1,30 @@
 ---
 uid: P000210
 name: $\alpha_1$
+aliases:
+- $\alpha_1$-space
 refs:
   - zb: "0774.54019"
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties. (Nyikos, P.)
   - zb: "0275.54004"
     name: The frequency spectrum of a topological space and the classification of spaces (Arkhangel’skii, A. V.)
+  - zb: "1348.54033"
+    name: Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)
 ---
 
-$X$ is an $\alpha_1$ space if for all $x \in X$ 
-and for all countable collections of sequences 
-$\gamma$ converging to $x$ there exists a sequence 
-$B$ converging to $x$ such that $A\setminus B$ is 
-finite for all $A \in \gamma$.  
+For every point $x \in X$ and every countable collection $\gamma$
+of (injective) sequences converging to $x$ there exists an (injective) sequence 
+$B$ converging to $x$ such that $A\setminus B$ is finite for each $A \in \gamma$.
+(One can always assume $B\subseteq\bigcup\gamma$.)
 
-The $\alpha_i$ properties for $i = 1, 2, 3, 4$ 
-are due to Arkhangel’skii ({{zb:0275.54004}}), 
-however the definition stated here is as presented 
-by Nyikos in {{zb:0774.54019}}. 
+PENDING ISSUE: equivalence with disjointness condition?
+
+Note: In the context of the definitions above, by "sequence" one has to understand an injective sequence,
+and we identify it with its range, which is a countably infinite set $A$.
+If one bijective enumeration of $A$ converges to $x$, all bijective enumerations of $A$ converge to $x$.
+Convergence thus becomes a property of countably infinite sets,
+with $A$ converging to $x$ if every neighborhood of $x$ contains all but finitely many points of $A$.
+
+The $\alpha_i$ properties for $i = 1, 2, 3, 4$ (sometimes called "sheaf amalgamation properties")
+are due to Arkhangel'skii ({{zb:0275.54004}}).
+The definition stated here is as presented by Nyikos in {{zb:0774.54019}}.

properties/P000211.md

@@ -1,21 +1,30 @@
 ---
 uid: P000211
 name: $\alpha_2$
+aliases:
+- $\alpha_2$-space
 refs:
   - zb: "0774.54019"
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties. (Nyikos, P.)
   - zb: "0275.54004"
-    name: The frequency spectrum of a topological space and the classification of spaces (Arkhangel’skii, A. V.)
+    name: The frequency spectrum of a topological space and the classification of spaces (Arkhangel'skii, A. V.)
+  - zb: "1348.54033"
+    name: Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)
 ---
 
-$X$ is an $\alpha_2$ space if for all $x \in X$ 
-and for all countable collections of sequences 
-$\gamma$ converging to $x$ there exists a sequence 
-$B$ converging to $x$ such that $A\cap B$ is infinite 
-for all $A \in \gamma$, or equivalently, if $A\cap B$ 
-is non-empty for all $A \in \gamma$.  
+For every point $x \in X$ and every countable collection $\gamma$
+of (injective) sequences converging to $x$ there exists an (injective) sequence 
+$B$ converging to $x$ such that $A\cap B$ is infinite for each $A \in \gamma$.
+(One can always assume $B\subseteq\bigcup\gamma$.)
 
-The $\alpha_i$ properties for $i = 1, 2, 3, 4$ 
-are due to Arkhangel’skii ({{zb:0275.54004}}), 
-however the definition stated here is as presented 
-by Nyikos in {{zb:0774.54019}}. 
+Equivalently, the same condition holds with the additional requirement that the sequences in $\gamma$ are pairwise disjoint. (The equivalence follows from Lemma 1.2 in {{0774.54019}}.)
+
+Note: In the context of the definitions above, by "sequence" one has to understand an injective sequence,
+and we identify it with its range, which is a countably infinite set $A$.
+If one bijective enumeration of $A$ converges to $x$, all bijective enumerations of $A$ converge to $x$.
+Convergence thus becomes a property of countably infinite sets,
+with $A$ converging to $x$ if every neighborhood of $x$ contains all but finitely many points of $A$.
+
+The $\alpha_i$ properties for $i = 1, 2, 3, 4$ (sometimes called "sheaf amalgamation properties")
+are due to Arkhangel'skii ({{zb:0275.54004}}).
+The definition stated here is as presented by Nyikos in {{zb:0774.54019}}.

properties/P000212.md

@@ -1,20 +1,30 @@
 ---
 uid: P000212
 name: $\alpha_3$
+aliases:
+- $\alpha_3$-space
 refs:
   - zb: "0774.54019"
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties. (Nyikos, P.)
   - zb: "0275.54004"
-    name: The frequency spectrum of a topological space and the classification of spaces (Arkhangel’skii, A. V.)
+    name: The frequency spectrum of a topological space and the classification of spaces (Arkhangel'skii, A. V.)
+  - zb: "1348.54033"
+    name: Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)
 ---
 
-$X$ is an $\alpha_3$ space if for all $x \in X$ 
-and for all countable collections of sequences 
-$\gamma$ converging to $x$ there exists a sequence 
-$B$ converging to $x$ such that $A\cap B$ is 
-infinite for infinitely many $A \in \gamma$.  
+For every point $x \in X$ and every countable collection $\gamma$
+of (injective) sequences converging to $x$ there exists an (injective) sequence 
+$B$ converging to $x$ such that $A\cap B$ is infinite for infinitely many $A\in\gamma$.
+(One can always assume $B\subseteq\bigcup\gamma$.)
 
-The $\alpha_i$ properties for $i = 1, 2, 3, 4$ 
-are due to Arkhangel’skii ({{zb:0275.54004}}), 
-however the definition stated here is as presented 
-by Nyikos in {{zb:0774.54019}}.  
+Equivalently, the same condition holds with the additional requirement that the sequences in $\gamma$ are pairwise disjoint. (The equivalence follows from Lemma 1.2 in {{0774.54019}}.)
+
+Note: In the context of the definitions above, by "sequence" one has to understand an injective sequence,
+and we identify it with its range, which is a countably infinite set $A$.
+If one bijective enumeration of $A$ converges to $x$, all bijective enumerations of $A$ converge to $x$.
+Convergence thus becomes a property of countably infinite sets,
+with $A$ converging to $x$ if every neighborhood of $x$ contains all but finitely many points of $A$.
+
+The $\alpha_i$ properties for $i = 1, 2, 3, 4$ (sometimes called "sheaf amalgamation properties")
+are due to Arkhangel'skii ({{zb:0275.54004}}).
+The definition stated here is as presented by Nyikos in {{zb:0774.54019}}.

properties/P000213.md

@@ -1,20 +1,30 @@
 ---
 uid: P000213
 name: $\alpha_4$
+aliases:
+- $\alpha_4$-space
 refs:
   - zb: "0774.54019"
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties. (Nyikos, P.)
   - zb: "0275.54004"
     name: The frequency spectrum of a topological space and the classification of spaces (Arkhangel’skii, A. V.)
+  - zb: "1348.54033"
+    name: Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)
 ---
 
-$X$ is an $\alpha_4$ space if for all $x \in X$ 
-and for all countable collections of sequences 
-$\gamma$ converging to $x$ there exists a sequence 
-$B$ converging to $x$ such that $A\cap B$ is nonempty 
-for infinitely many $A \in \gamma$.  
+For every point $x \in X$ and every countable collection $\gamma$
+of (injective) sequences converging to $x$ there exists an (injective) sequence 
+$B$ converging to $x$ such that $A\cap B$ is nonempty for infinitely many $A\in\gamma$.
+(One can always assume $B\subseteq\bigcup\gamma$.)
 
-The $\alpha_i$ properties for $i = 1, 2, 3, 4$ 
-are due to Arkhangel’skii ({{zb:0275.54004}}), 
-however the definition stated here is as presented 
-by Nyikos in {{zb:0774.54019}}. 
+Equivalently, the same condition holds with the additional requirement that the sequences in $\gamma$ are pairwise disjoint. (The equivalence follows from Lemma 1.2 in {{0774.54019}}.)
+
+Note: In the context of the definitions above, by "sequence" one has to understand an injective sequence,
+and we identify it with its range, which is a countably infinite set $A$.
+If one bijective enumeration of $A$ converges to $x$, all bijective enumerations of $A$ converge to $x$.
+Convergence thus becomes a property of countably infinite sets,
+with $A$ converging to $x$ if every neighborhood of $x$ contains all but finitely many points of $A$.
+
+The $\alpha_i$ properties for $i = 1, 2, 3, 4$ (sometimes called "sheaf amalgamation properties")
+are due to Arkhangel'skii ({{zb:0275.54004}}).
+The definition stated here is as presented by Nyikos in {{zb:0774.54019}}.

theorems/T000733.md

@@ -0,0 +1,12 @@
+---
+uid: T000733
+if:
+  P000210: true
+then:
+  P000211: true
+refs:
+  - zb: "1348.54033"
+    name: Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)
+---
+
+Follows directly from the definitions.

theorems/T000734.md

@@ -0,0 +1,12 @@
+---
+uid: T000734
+if:
+  P000211: true
+then:
+  P000212: true
+refs:
+  - zb: "1348.54033"
+    name: Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)
+---
+
+Follows directly from the definitions.

theorems/T000735.md

@@ -0,0 +1,12 @@
+---
+uid: T000735
+if:
+  P000212: true
+then:
+  P000213: true
+refs:
+  - zb: "1348.54033"
+    name: Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)
+---
+
+Follows directly from the definitions.