# Separation of compacts sets in a LCHS

Could someone help me with the following question?

Let $$X$$ be a locally compact Hausdorff space (LCHS) and $$K_1,K_2\subset{X}$$ two disjoint compact sets. Then there exist $$V_1,V_2$$ disjoint open sets such that $$K_1\subset{V_1},K_2\subset{V_2}$$ and $$\overline{V_1},\overline{V_2}$$ are compact.

I have found several proofs of this fact (see Lemma 5.1 here), but they use sophisticated arguments.

Thanks.

• – mathlife Mar 17 at 10:14
• I am looking for a simpler proof, if it exists. – mathlife Mar 17 at 10:15

Let K,L be two disjoint compact sets.

Pick y in L.
For all x in K, exists open disjoint U$$_x$$, V$$_x$$ with x in U$$_x$$, y in V$$_x$$.
C = { U$$_x$$ : x in K } covers K.
Let Cf be a finite subcover of C.
U = $$\cup$${ U$$_x$$ : U$$_x$$ in Cf }, V = $$\cap$${ V$$_x$$ : V$$_x$$ in Cf } are open.
K subset U, y in V and U and V are disjoint.

Using the above, for all y in L,
exists open disjoint U$$_y$$, V$$_y$$ with K subset U$$_y$$, y in V$$_y$$.
C = { V$$_y$$ : y in L } covers L.
Let Cf be a finite subcover of C.
U = $$\cap$${ U$$_x$$ : U$$_x$$ in Cf }, V = $$\cup$${ V$$_x$$ : V$$_x$$ in Cf } are open.
K subset U, L subset V and U and V are disjoint.

Thus two disjoint compact set of a Hausdorff space can be separated by disjoint open sets.

Assume K is compact and U is open within a locally compact space Hausdorff space.
If K subset U, then for all x in K,
exists open U$$_x$$ with x in U$$_x$$ subset U, compact $$\overline U_x$$.
C = { U$$_x$$ : x in K } covers K.
Let Cf be a finite subcover of C.
U' = $$\cup$${ U$$_x$$ : U$$_x$$ in Cf } is open. K subset U' subset U.
Finally show $$\overline {U'}$$ is compact.

Do the same for L and pull together the loose ends for the conclusion you wish to prove. Lengthy but mostly simple stuff.

• I don't undertsand this sentence of the second part: "If $K$ subset $U$, then for all $x$ in $K$ exists open $U_x$ with $x$ in $U_x$ subset U, compact $\overline{U_x}$ " – mathlife Mar 17 at 17:33
• @mathlife. Space is locally compact. – William Elliot Mar 17 at 22:03