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-homemorphic
for a judicious choice of X, T, T', and T''.

Gerhard "Ask Me About System Design" Paseman, 2010.07.05