The following lemma about locally compact (but not necessarily Hausdorff) spaces or continuous lattices appears frequently but without citation. It is easy to prove but important in proofs.

If a compact subspace is covered by two open subspaces, $K\subset U_1\cup U_2$, then there are compact subspaces $L_1$ and $L_2$ and open ones $V_1$ and $V_2$ such that $K\subset V_1\cup V_2$, $V_1\subset L_1\subset U_1$ and $V_2\subset L_2\subset U_2$.

The earliest occurrence that I know is in *Adjoint Product and Hom Functors in General Topology* by Peter Wilker in *Pacific Journal of Mathematics* **34**(1970)269--283.  This paper was part of the development of [giving ${\mathbf{Top}}(X,Y)$ an approprate topology](http://mathoverflow.net/questions/127841/giving-topx-y-an-appropriate-topology/127853#127853).

Was this property identified any earlier than 1970?


  [1]: http://mathoverflow.net/questions/127841/giving-topx-y-an-appropriate-topology/127853#127853