Construct an example proving that a continuous image of a second countable space may be not second countable.
I construct an example by taking two different topology on $I=[0,1]$, $(I,\mathcal{X})$ and $(I,\mathcal{Y})$ where $\mathcal{X}$ is the standard one, and $\mathcal{Y}$ is generated from the base of the open interval with end points in Cantor set. The map is the identity map.
Is my construction right? Is there any other constructions?
Added: My construction is wrong...