Add proof for T28 (#1710)

Author: felixpernegger

theorems/T000028.md

@@ -7,10 +7,11 @@ if:
   - P000028: true
 then:
   P000005: true
-refs:
-- zb: "0386.54001"
-  name: Counterexamples in Topology
 ---
 
-
-Asserted in Figure 7 of {{zb:0386.54001}}.
+Suppose we have a closed set $A$ and a point $x\notin A$.
+Let $(U_n)_{n\in \mathbb{N}}$ be a countable neighbourhood basis for $x$.
+Since $X$ is {P3}, the (countably many) open sets $X\setminus\overline{U_n}$ cover $A$.
+Since $A$ is closed in $X$, it is also {P19};
+consequently $A\subseteq\bigcup_{n\in I}(X\setminus\overline{U_n})$ for some finite $I\subseteq\mathbb N$.
+Taking complements, one sees that $\bigcap_{n\in I}\overline{U_n}$, which is a closed neighborhood of $x$, is disjoint from $A$.