Add alpha_i properties (#1366)

Author: NAndrews4122

properties/P000210.md

@@ -0,0 +1,20 @@
+---
+uid: P000210
+name: $\alpha_1$
+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.)
+---
+
+$X$ 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 $A \in \gamma$.  
+
+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}}. 

properties/P000211.md

@@ -0,0 +1,21 @@
+---
+uid: P000211
+name: $\alpha_2$
+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.)
+---
+
+$X$ 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 $A \in \gamma$.  
+
+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}}. 

properties/P000212.md

@@ -0,0 +1,20 @@
+---
+uid: P000212
+name: $\alpha_3$
+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.)
+---
+
+$X$ 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 $A \in \gamma$.  
+
+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}}.  

properties/P000213.md

@@ -0,0 +1,20 @@
+---
+uid: P000213
+name: $\alpha_4$
+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.)
+---
+
+$X$ 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 $A \in \gamma$.  
+
+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}}.