Continuous, Invertible Bijection Which is Not a Homeomorphism

The definition of a topology homeomorphism requires it to be continuous, invertible, and to have a continuous inverse. This note gives an example of a function which satisfies the first two properties but not the third.

Example

Let

X={0,1}andY={2,3}

be topological spaces with corresponding topologies

TX=P(X)andTY={∅,X,{2}}.

Then define f:X→F by f(0)=2 and f(1)=3. f is a continuous bijection however g=f−1 is not continuous.

This function is trivially continuous because the input space topology is the discrete topology, so all pre-images are open.

However g−1({1})={3} which is not open.

With some additional conditions on the input and output space, the above behaviour can be excluded. Namely when the domain is a compact space and the codomain is a Hausdorff space.