パラコンパクト空間
We will show that there is closed set C ⊃ W U {\displaystyle C\supset W_{U}} such that x ∉ C {\displaystyle x\notin C} (this means simply x ∉ W U ̄ {\displaystyle x\notin {\bar {W_{U}}}} by definition of closure). Since we chose V {\displaystyle {\mathcal {V}}} to be locally finite there is neighbourhood V [ x ] {\displaystyle V[x]} of x {\displaystyle x} such that only finitely many sets U 1 , . . . , U n ∈ { A ∈ V : A ̄ ⊆ U } {\displaystyle U_{1},...,U_{n}\in \{A\in {\mathcal {V}}:{\bar {A}}\subseteq U\}} have non-empty intersection with V [ x ] {\displaystyle V[x]} . We take their closures U 1 ̄ , . . . , U n ̄ {\displaystyle {\bar {U_{1}}},...,{\bar {U_{n}}}} and then V := V [ x ] ∖ ∪ U i ̄ {\displaystyle V:=V[x]\setminus \cup {\bar {U_{i}}}} is an open set (since sum is finite) such that V ∩ W U = ∅ {\displaystyle V\cap W_{U}=\varnothing } .