By Fox R.H. (ed.), Spencer D.C. (ed.), Tucker A.W. (ed.)

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Such thoughts may well have influenced Lefschetz; for he undertook an intensive investigation of local -connectedness. It is not easy to communicate in a few words the spirit of the time, that is, the influential ideas, and the problems considered to be im- One outstanding problem was (and still is) the extension to of the topological characterizations of the 1-cell dimensions higher and 2-cell. Local connectedness in the sense of point-set topology had portant. an important role in these characterizations.

Lefschetz showed [58] LC-space, for ANR that the class of compact metric LC-spaces coincides with the spaces of Borsuk. Lefschetz defined an even broader concept of local-connectedness by HLC? HLO, etc. He simply any g-cycle in U bounds a chain of compact metric HLC spaces enjoys in the sense of homology [64], denoted modified the above definition to read , ' : F He showed that the class many of the properties of complexes. in '. theorem is valid for such a space In particular the fixed-point [7].

He axiomatized the existence of such concept of a cochain cup-product; and he proved classes using the products and uniqueness of induced products of acyclic-carrier type of argument. most conceptual. Whitney's method was the most general and the His cochain products were likewise not necessarily associative. 2) in the subdivision. NORMAN 40 The three methods were STEENROD E. alike in that they involved constructions entirely within the initial complex. Lefschetz's procedure was still more general; but entirely in keeping with methods he had used earlier.