# Math Help - Countability in topological spaces

1. ## Countability in topological spaces

Hi,

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

Can someone explain this for me?

Thanks!

2. ## Re: Countability in topological spaces

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