What are nice examples of topological spaces $X$ and $Y$ such that $X$ and $Y$ are **not** homeomorphic but there do exist continuous bijections $f: X \to Y$ and $g: Y \to X$?



Non-homeomorphic spaces that have continuous bijections between them
Recycling an old (ca. 1998) sci.math post:


" Anyone know an example of two topological spaces $X$ and $Y$ 
 with continuous bijections $f:X\to Y$ and $g:Y\to X$ such that 
 $f$ and $g$ are not homeomorphisms? 


Let $X = Y = Z \times \{0,1\}$ 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 $f:X\to Y$ and $g:Y\to X$ 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


