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 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.
I took this out of a study book from University.  This book was written by me and by a handful of other high-performing students, edited and checked by our professors and other faculty.  Its basically an academia-verified study guide written by the best of students, checked by the best of teachers, spanning all of high level mathematics. Content for this book was taken directly out of college text books, lecture notes, etc.  The book was written for the purposes of studying, review, and refreshing over the years in academia.  It comprises a multitude of high level courses in abstract mathematics spanning several years of the bachelors program, from advanced logic to topology, set theory to abstract algebra. Its several hundred pages long, full of definitions, theorems, etc., without any of the explanatory filler one normally finds in lecture notes or text books.  Familiarity with the subject matter is assumed, as its meant only for review. And its a work in progress.

Content was made as abstract as possible: No assumption is made about the types of vectors or fields or sets being employed.  Nor the operators.  Thus no numbers are used. We refer to generic sets, generic identity elements, generic order relations, generic operators, etc.

Yes, these clipped images from a PDF were written by me and others in LaTeX, and verified by experts in the fields. No cheating here. This is my content, as good as if I typed it out into stack exchange myself.  If there are errors or typeos, or anything you conceptually disagree with on a well-researched and mathematically rigorous basis, please do let me know. Im open to being corrected, just not resentful egotists who want to play petty games. Otherwise I cant imagine why the inevitable down votes would be justified, but I doubt those of you who would even read this far.



On the concept of metrics







On the concept of a topology








Combining the two