(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.