TITLE: Are Disjoint Open Subsets Contained in Disjoint Open Sets in a General Topological Space?
QUESTION [7 upvotes]: This was a question that I had while revising metric spaces.
For a metric space $X$, let $A\subset X$. We know that if $G, H$ are disjoint open sets in $A$, $\exists$ disjoint open sets $U, V \subset X$ such that $G = U \cap A$, $H = V \cap A$.
What I was wondering is that if we can generalize it to general topological spaces, that is, for a topological space $X$ and $A$ having subspace topology. 
Also, if this does not hold for a general topological space, what restrictions (like Hausdorff-ness) do we need to place for this to hold?

REPLY [2 votes]: Minimal counterexample. Take $X = \{0,1,2\}$ with the topology $\{\emptyset, \{0\}, \{0,1\}, \{0,2\}, \{0,1,2\}\}$, $A = \{1,2\}$ $G = \{1\}$ and $H = \{2\}$. Then $G$ and $H$ are disjoint open subsets of $A$. However, every open set of $X$ containing either $U$ or $V$ contains $0$. Thus the intersection of any pair of candidates for $U$ and $V$ will contain $0$.