Pseudo metric spaces are not Hausdorff. I know that every metric space is Hausdorff,and every metric space is Pseudo metric.
How can I prove that every Pseudo metric space is not Hausdorff??
 A: A pseudometric satisfies all of the requirements of a metric except one: it need not separate points. Suppose that $d$ is a pseudometric on $X$ that is not a metric; then there must be points $x,y\in X$ such that $d(x,y)=0$, but $x\ne y$. That’s the only way that a pseudometric can fail to be a metric. 
Now suppose that $x\in U$, where $U$ is an open sets in $X$. Then by the definition of the pseudometric topology there is a real number $r>0$ such that $B(x,r)\subseteq U$, where as usual $$B(x,r)=\big\{z\in X:d(x,z)<r\big\}\;.$$ But $d(x,y)=0<r$, so $y\in B(x,r)\subseteq U$. In other words, every open neighborhood of $x$ also contains $y$, and we cannot possibly find open sets $U$ and $V$ such that $x\in U$, $y\in V$, and $U\cap B=\varnothing$: no matter what $U$ and $V$ pick, it will always be the case that $y\in U\cap V$. (It will also be true that $x\in U\cap V$.)
You might find it useful to recognize that the pseudometric $d$ induces an equivalence relation $\sim$ on $X$: for $x,y\in X$ put $x\sim y$ iff $d(x,y)=0$. I’ll leave to you the easy proof that $\sim$ is an equivalence relation and that its equivalence classes are closed sets. The pseudometric $d$ treats the points in a single equivalence classes as if they were identical: if $x\sim y$ and $u\sim v$, then $d(x,u)=d(y,v)$. From the standpoint of $d$, $x$ and $y$ are interchangeable, as are $u$ and $v$. 
I don’t know whether you’ve studied quotient topologies at all yet, but when you do, you can check that the quotient space $X/\sim$ whose points are the $\sim$-equivalence classes actually becomes a metric space with a metric $\hat d$ when you define $\hat d\big([x],[y]\big)=d(x,y)$ (where $[x]$ and $[y]$ are the equivalence classes of $x$ and $y$, respectively). It’s what you get if you simply pretend that all of the points that are $d$-distance $0$ from one another are really the same point.
