Strengthen T552

theorems/T000552.md

@@ -2,12 +2,10 @@
 uid: T000552
 if:
  and:
- - P000196: true
- - P000016: true
+ - P000146: true
+ - P000036: true
  - P000086: true
 then:
  P000129: true
 ---
-If a {P196} space is {P16}, then it must have a second largest open set 
-(otherwise, $\mathcal T_X\setminus\{X\}$ would be totally ordered with no upper bound and therefore an open cover with no finite subcover).
-To be {P86}, no points can lie in this second-largest set, so the space is {P129}.
+Any clopen partition of a {P36} space $X$ must contain $X$, so to admit clopen refinements every open cover must contain $X$. Thus, the union of all open sets except for $X$ cannot equal $X$ (as that would be an open cover not containing $X$), so some points have $X$ as their only neighborhood. By homogeneity, this must then be true of all points, i.e. the space is {P129}.