Is the definition of metric topology, discrete topology and topology equivalent? I am taking a course in intro topology and very confused because the lecturer has only given us a definition of metric topology while all the texts I have read have the axiomatic definition of topology.
I don't think they are the samee but what is the difference? Also is metric topology same as 'discrete topology'?
Definitions I have are: 
Metric topology: Collection of open sets such that all subsets are in X.
Topology: The three axioms of it should contain X, empty set, union and intersectoin should be in the topology.
Discrete topology: T is the collection of all subsets of a non empty set X. 
To me discrete and metric look the same?
I am not understanding anything in lectures so I am trying to teach myself with resources online. Thanks for your help.
 A: A metric $d$ on $X$ lets us define a notion of open sets, i.e. a topology $\mathcal{T}_d$. The latter obeys the abstract axioms of a topology.
But many abstract topologies cannot have a metric defined on them that defines that topology in this way. There are theorems on when this is and is not the case. Many topologies in practical maths arise from metrics, so in the beginning there is often more focus on metrics (analysis has a lot of use for it). E.g. at my university we had a course in "metric topology" in year 1, introductory general topology in year 2, advanced topics later. 
The discrete topology is actually a general topology that is generated by a metric: $d(x,y)=1$ when $x \neq y$, $d(x,x)=0$. The usual topology on $\Bbb R$, $\Bbb Q$ etc. all arise from metrics. 
A: A topology $\tau$ on a set $X$ is a collection of subsets of $X$ satisfying three conditions (and you know what conditions they are).
In a metric space $(X,d)$, we can define open balls $B_r(x)$ to be
$$B_r(x)=\{y\in X: d(x,y)<r\}.$$
Now the collection of all subsets of $X$ which are called open balls, $\{B_r(x)\}$, satisfy the three conditions of being a topology (in fact a basis for a topology) and this topology is called "metric topology" or "the topology induced from a metric".
Note that not all topologies induced from a metric, e.g the trivial topology $\tau=\{\varnothing, X\}$.
We also have phrases like "partition topology" which a partition on a set is given and the subsets of the partition satisfy the three conditions.
