Orientation on $\mathbb{CP}^2$ I am confused by the orientation of a topological manifold.
My understanding is: An orientation of a topological manifold is a choice of generator of the $H^n(M,\mathbb Z)$. So given a manifold, we could have 2 orientation defined on the manifold. For $\mathbb{CP}^2$ on orientation is determined by the complex structure, the other orientation is denoted by $\overline{\mathbb{CP}}^2$. And it is well known that there is no orientation reversing map from $\mathbb{CP}^2$ to itself. 
My question is: are they homeomorphic? I guess they are not homeomorphic.
But I am confuesed, doesn't the orientations defined on the same manifold, how come after reversing the orientation they become not homeomorphic?

A topological manifold is a topological space with extra conditions. In particular, for $\mathbb{CP}^2$, whatever the orientation you choose, the underlying topological space is the same. Therefore, the identity map is a homeomorphism.
However, they are not equivalent as oriented topological manifold, since there exists no orientation-reversing map from $\mathbb{CP}^2$ to itself. Hence, I think the confusion arises from a confusion with terminology.
