continuous Function In Metric/Topology Spaces Let $X,Y$ be a metric spaces then there is a theorem 
$f$ is continuous iff the pre-image of an open set is an open set
On the other hand in topology spaces it seems that the definition is:
Let $X,Y$ be a topology spaces and $f:X\to Y$ then 
$f$ is continuous if the pre-image of an open set is an open set
So in metric spaces it is $\iff$ and in topology spaces just in one direction? 
 A: 
So in metric spaces it is $\iff$ and in topology spaces just in one direction? 

The metric world has its definition and so does the topological world. It is a theorem that they are equivalent. But this is only since we can generate a topology from a metric but not the other way around. So in a way metric spaces are topological spaces (not literally but they induce a topology) but topological spaces need not be metric in general (in the sense that not every topology arises from a metric).
More generally: there are lots of continuity definitions: Cauchy, uniform, Holder, sequential, $\epsilon-\delta$, Lipshitz, via nets, via preimages, etc. etc. Some of them are equivalent, some are not. Some are only equivalent under the Axiom of Choice.
For metric spaces I assume this is the definition you are referring to: $f:X\to Y$ is continuous if for any $x\in X$ and any $\epsilon\in\mathbb{R}, \epsilon>0$ there exists $\delta\in\mathbb{R}, \delta >0$ such that if $y\in X$ with $d_X(x,y)<\delta$ then $d_Y(f(x),f(y))<\epsilon$. Also known as $\epsilon-\delta$ definition.
Note that this definition does not make sense in the topological world where we don't have the notion of distance.
For topological spaces the definition is a bit shorter: $f:X\to Y$ is continuous if $f^{-1}(U)$ is open whenever $U\subseteq Y$ is.
So these are definitions. And it is a theorem that in the metric world these two definitions are equivalent if we take the usual topology generated by a metric.
But there is at least one more important definition of continuity in topological spaces: $f:X\to Y$ is continuous if for any convergent net $(x_\alpha)\subseteq X$ and its limit point $x\in X$ we have that $f(x_\alpha)$ is a net convergent to $f(x)\in Y$.
So as you can see it is not unusual to have multiple definitions for things.
