(A paradox which arose with Yining Xia 10/95; not posted or mailed.) Distinction between direct sums and direct products -OR- Example showing the need to define topologies carefully Let A and B be two countable sets of earrings in S^3 which interlock in pairs. That is, each of A and B is a countable collection of circles which meet only at the north (resp. south) pole. The n-th circle of A is linked only with the n-th circle of B. The specific model we use takes A to consist of a family of coplanar parallel lines in R^3, say R^2 x Z inside R^3. Here we view R^3 as S^3 - {North}. Then B is a set of coplanar circles, each wrapping around one line, and each extended by a ray to a fixed point in this plane but far from the first plane. Clearly S^3 - A is the complement of the lines in R^3, which has B as a deformation retract. On the other hand, the descriptions of A and B inside S^3 are complementary, so S^3 - A and A have the same homotopy type, and thus the same homology and cohomology. Now, as A is a wedge of circles, H_1(A) is a direct sum of a countable number of copies of Z; then H_1(S^3-A) is the same. By Alexander duality, this last group is the same as the (Cech) cohomology of A, which is the direct product of a countable number of copies of Z. But Sum is not Product! Resolution of the paradox: need to look carefully at topologies involved. Viewed as subsets of S^3, A and B are _not_ infinite wedges of circles, in that every nbhd of the common point includes almost all of the circles in S^3. Computing (^-H)^1(A) as lim(->) of (^-H)^1(N), ranging over open neighborhoods N of A in S^3, we see (^-H)^1(A) is indeed a direct sum of copies of Z.