TITLE: A continuous image of a second countable space which is not second countable.
QUESTION [10 upvotes]: 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...

REPLY [2 votes]: This is exercise 16B.1 in Willard's General Topology: For each $n\in \Bbb N$, let $I_n$ be a copy of $[0,1]$. Let $X$ be the  disjoint union of the spaces $I_n$. Identify the left hand endpoints of all the $I_n$ and let $Z$ be the resulting quotient space. The distinguished point in $Z$ has no countable nhood base, though $X$ is second countable.