Countability in topological spaces

Hi,

I need to prove the following:

Consider the topological space $\displaystyle \mathbb{R}$ with the euclidian topology. Define an equivalence relation on $\displaystyle \mathbb{R}$ as follows $\displaystyle xRy \Leftrightarrow x=y \ \mbox{or} \ \{x,y\} \subset \mathbb{N}$. Now, consider the quotient topology on $\displaystyle \mathbb{R}/ R$ then this space is neither $\displaystyle \mbox{C}1$(first countable axiom) nor $\displaystyle \mbox{C}2$ (second countable axiom), because there doens't exist a countable base for the neighbourhoud filter of $\displaystyle \overline{0}$.

Can someone explain this for me?

Thanks!

Re: Countability in topological spaces

How to prove,in topological space,every compact space is separable.?