Frechet Urysohn + Weakly First Countable + US => First countable

properties/P000104.md

@@ -26,3 +26,5 @@ Defined in {{doi:10.2991/978-94-6239-216-8}}. Note that some sources, e.g., {{do
 #### Meta-properties
 
 - This property is not hereditary. (Example: {S156|P104} and {S23|P104}.) Compare with {P102}.
+- This property is hereditary with respect to open sets.
+- This property is hereditary with respect to closed sets.

properties/P000228.md

@@ -21,5 +21,7 @@ See Definition 2.3 in {{zb:0171.43603}}. This property is called *g-first counta
 ----
 #### Meta-properties
 
+- This property is hereditary with respect to open sets.
+- This property is hereditary with respect to closed sets.
 - $X$ satisfies this property if its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is preserved by arbitrary disjoint unions.

theorems/T000845.md

@@ -0,0 +1,20 @@
+---
+uid: T000845
+if:
+  and:
+  - P000228: true
+  - P000080: true
+  - P000099: true
+then:
+  P000028: true
+refs:
+  - zb: "0285.54022"
+    name: On defining a space by a weak base. (Siwiec)
+---
+
+See Theorem 1.10 in {{zb:0285.54022}}.
+There, the author assumes {P3} instead of {P99} (see section 1.7), but the latter is sufficient for the proof. {P3} is only used to show that if $x_1,x_2,\dots$ is a sequence converging to $x$, then $\{x,x_1,x_2,\dots\}$ is closed in $X$. But indeed, one can use the following general fact:
+
+"If $x_1,x_2,\dots$ is a sequence in a {P99} space $X$ converging to a point $x\in X$, then the set $\{x,x_1,x_2,\dots\}$ is sequentially closed."
+
+In combination with {T840}, the assertion follows.