Definition 3

A homeomorphism of a topological space $X$ onto a topological space $X'$ is an isomorphism of the topological structure of $X$ onto that of $X'$; that is to say, in accordance with the general definitions a bijection of $X$ onto $X'$ which transforms the set of open subsets of $X$ into the set of open subsets of $X'$.

$X$ and $X'$ are said to be homeomorphic if there is a homeomorphism of $X$ onto $X'$.