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It was in the presence of these cross-currents that Lefschetz made A space X his contribution to the theory of local-connectedness [57]. locally-connected in dimension q, if for each point x of X, and each neighborhood F of x, there is a neighborhood U of x such that each continuous map of a g-sphere into U can be contracted to a point in F. This property is abbreviated by the symbols LC A space is is LC n if it is LCQ for q = 0, 5 1, . . , n\ and it is LC if it is LCa . for all q. An which the neighborhood U corresponding to F can be chosen independently of q is called an LC-space.

3, pp. 63-77. BIBLIOGRAPHY 48 The role of algebra in topology. Bull. Amer. Math. , 43: 345-359. Address of retiring president of the American Mathematical Society. [71] On the fixed point formula. Ann. of Math. (2), 38: 819-822. [70] 1938 [72] Lectures on Algebraic Geometry (Part II) 1937-1938. Princeton Uni- versity Press. , planographed. [73] On chains of topological spaces. Ann. of Math. (2), 39: 383-396. [74] On locally connected sets and retracts. Proc. Nat. Acad. Sci. , 24: 392-393. [75] [76] Sur les transformations des complexes en spheres (note complementaire).

2) . 1), /i point of/. 2) reduces to the previous formula for L(f). If . 5. Consider now The dim N < dim M The graphs of/ and g are M x N has dimension m + n < 2m. The geometric the case w-cycles, but intersection of the still intersection ring . e. a a with Kronecker index. If the two cycles are in general 0-cycle the dimension of the intersection ism n. position, It is quite probable that this consideration was one of those which led Lefschetz to develop the theory of the intersection ring of a manifold [31,32,33].