Recycling an old (ca. 1998) sci.math post:
": Anyone know an example of two topological spaces X and Y : with continuous bijections f:X-->Y and g:Y-->X such that : f and g are not homeomorphisms?
Let X = Y = Z x {0,1} as sets, where Z is the set of integers. We declare that the following subsets of X are open for each n>0. {(-n,0)} {(-n,1)} {(0,0)} {(0,0),(0,1)} {(n,0),(n,1)} This is a basis for a topology on X.
We declare that the following subsets of Y are open for each n>0. {(-n,0)} {(-n,1)} {(0,0),(0,1)} {(n,0),(n,1)} This is a basis for a toplogy on Y.
Define f:X-->Y and g:Y-->X by f((n,i))=(n,i) and g((n,i))=(n+1,i). Then f and g are continuous bijections, but X and Y are not homeomorphic.
This example is due to G. Paseman.
David Radcliffe "
More generally, take a space X with three successively finer topologies T, T' and T''. Form two spaces which have underlying set ZxX, and "form the infinite sequences" .... T T T T' T'' T'' T'' .... and ... T T T T T'' T'' T'' T'' .... The continuous maps will take a finer topology in one sequence to a rougher topology in the other. You can make them bijective, and show that they are obviously non-homeomorphic for a judicious choice of X, T, T', and T''.
Gerhard "Ask Me About System Design" Paseman, 2010.07.05