Example

If $X$ and $X'$ are two discrete spaces, any bijection of $X$ onto $X'$ is a homeomorphism.

The following criterion follows immediately from the definition of a homeomorphism: for a bijection $f$ of a topological space $X$ onto a topological space $X'$ to be a homeomorphism, it is necessary and sufficient that the image under $f$ of each open set in $X$ is an open set in $X'$, and that the inverse image under $f$ of each open set in $X'$ is an open set in $X$.