Open Sets: A Real Life Example Okay so I get the grasp of a Topological Space, and that the elements in a Topology are open sets. I also understand that there are many different Topologies even on the same space. 
What are some real life examples of a Topology? Also what are some examples of a real life scenario of two different Topologies on the same set?
I guess what I don't fully grasp is the purpose Topology can play in the real world. Not to say that it isn't important.  
 A: It's important to understand that when talking about topology open set means a set what belongs to the topology. You should't think about it as something 'open' in any other sense.
So of course the most common example is:
1) All open (in traditional sense) subsets of $R^n$ form a topology over $R^n$
There are trivial examples like:
2) For any set, all subsets form a topology. Then by definition all subsets are both open and closed.
There are also harder to graps topologies, like:
3) Zariski topology is the topology for which closed sets are all subsets of, which are zeroes of some polynomial.
For the real line this means that all finite sets of points are closed. This is also called "finite complement topology".
4) On the real line there is another topology called "lower limit topology". It is defined such that open sets are all half open intervals $[a, b)$ (and therefore all their unions).
It is a useful exercise to prove that all those examples are indeed topologies, i.e. they satisfy the definitions.
