Prove a topological space is normal if and only if for each pair of open sets $U,V$ such that $U cup V=X$, there exists open sets $U’$ and $V’$ such that $overline U’ subset U$ and $overline V’ subset V$ and $U’ cup V’=X$
$(Rightarrow)$ Since $U cup V=X implies (X-U) cap (X-V)=varnothing$. $X-U$ and $X-V$ are disjoint closed sets. Then by normality of $X$, there are disjoint open sets $W,Z$ with $X-U subset W$ and $X-V subset Z$, and an equivalent definition of normality states there are open sets $W’,Z’$ containing $X-U$ and $X-V$ with $overline W’ subset W$ and $overline Z’ subset Z$. Now I am unsure how to show that $W’ cup Z’=X$, which is what I need to show in the problem.
$(Leftarrow)$ Suppose the second part of the statement holds. Let $A$ be closed in $X$ and $U$ be any open set containing $X$. Then the existence of an open set $U’$ containing $A$ such that $overline U’ subset U$ implies $X$ is normal.
Comment:I am having difficulty understanding (why/if) $W’ cup Z’=X$ holds in the first direction. Also I am skeptical whether I started the second direction correctly by starting with a closed set, and an arbitrary open set containing this closed set. I also realize it may not be true that $U’$ contains the closed set $A$, which adds to my skepticism. Any ideas?
For smooth projective algebraic curves over finite fields. Do they admit a finite Galois cover of any degree, that is not induced by just base extension?
What are the scenes being portrayed on the covers of the Player’s Handbook (PHB), Dungeon Master’s Guide (DMG) and Monster Manual (MM) and what monsters do the scenes contain?
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