gn.general topology - 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 $f:Xto Y$ and $g:Yto X$ 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 $f:Xto Y$ and $g:Yto X$ 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

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