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.
Was this property identified any earlier than 1970?
The definition of locally compact that I use is that if $x\in U$ then there are $K$ and $V$ with $x\in V\subset K\subset U$ (with similar letter conventions as above). This definition was stated in Local Compactness and Continuous Lattices by Karl Hofmann and Michael Mislove in Continuous Lattices edited by Bernhard Banaschewski and Rudolf-Eberhard Hoffmann (NB different spelling), Springer Lecture Notes in Mathematics, 871 (1981) 209--248.
However, the Wilker paper is not specifically about local compactness in this sense, even though he anticipates many of the ideas of continuous lattices and the Scott topology.
The context in which I ask the question is a draft paper of mine entitled Local Compactness and Bases in various formulations of Topology.
