pi-base · felixpernegger · May 19, 2026 · May 19, 2026 · May 21, 2026 · May 22, 2026
diff --git a/theorems/T000895.md b/theorems/T000895.md
@@ -0,0 +1,13 @@
+---
+uid: T000895
+if:
+  P000174: true
+then:
+  P000210: true
+---
+
+To prove the result it is enough to show every countable subspace of $X$ is {P210}.
+Note that each subspace of $X$ is also {P174}.
+So wlog we may assume $X$ is {P57}, and hence {P28}
+[(Explore)](https://topology.pi-base.org/spaces?q=Countable%2BWell-based%2B%7EFirst+countable).
+And {T748}.
diff --git a/theorems/T000896.md b/theorems/T000896.md
@@ -0,0 +1,9 @@
+---
+uid: T000896
+if:
+  P000147: true
+then:
+  P000210: true
+---
+
+Utilising the metaproperties of {P210} and {P147}, we may assume that $X$ is {P57}. But then the assertion follows from {T238}, {T748}, {T894} and {T285}.
-Utilising the metaproperties of {P210} and {P147}, we may assume that $X$ is {P57}. But then the assertion follows from {T238}, {T748}, {T894} and {T285}.
+To prove the result it is enough to show every countable subspace of $X$ is {P210}.
+Note that each subspace of $X$ is also {P147}.
+So wlog we may assume $X$ is {P57}, and hence {P28}
+[(Explore)](https://topology.pi-base.org/spaces?q=Countable%2BP-space%2B%7EFirst+countable).
+And {T748}.
-Utilising the metaproperties of {P210} and {P147}, we may assume that $X$ is {P57}. But then the assertion follows from {T238}, {T748}, {T894} and {T285}.
+To prove the result it is enough to show every countable subspace of $X$ is {P210}.
+Note that each subspace of $X$ is also {P147}.
+So wlog we may assume $X$ is {P57}, and hence {P28}
+[(Explore)](https://topology.pi-base.org/spaces?q=Countable%2BP-space%2B%7EFirst+countable).
+And {T748}.
diff --git a/theorems/T000897.md b/theorems/T000897.md
@@ -0,0 +1,13 @@
+---
+uid: T000897
+if:
+  P000187: true
+then:
+  P000210: true
+---
+
+To prove the result it is enough to show every countable subspace of $X$ is {P210}.
+Note that each subspace of $X$ is also {P187}.
+So wlog we may assume $X$ is {P57}, and hence {P28}
+[(Explore)](https://topology.pi-base.org/spaces?q=Countable%2BW-space%2B%7EFirst+countable).
+And {T748}.