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

\[ X = \{0, 1\} \quad \text{and} \quad Y = \{2, 3\}\]

be topological spaces with corresponding topologies

\[ \mathcal{T}_X = \mathcal{P}(X) \quad \text{and} \quad \mathcal{T}_Y = \{\varnothing, X, \{2\}\}.\]

Then define \(f : X \to F\) 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.