In order to establish an Alexander duality theorem for compact subsets of $S^n$, Steenrod introduced in 1940 a new type of homology of metric compacta; a summary of the construction is given in the entry, Steenrod homology. The same problem led K. A. Sitnikov in 1951 to an equivalent theory and in 1960 J. Milnor gave an axiomatic characterization of the resulting ‘Steenrod-Sitnikov homology’.

Several authors extended this to compact Hausdorff spaces and then Lisica and Mardešić gave a definition for arbitrary spaces.