(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.