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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.

TeeJ Lockwood
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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.

Petar Ivanov
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  • Thank you for your examples for Topologies! However I am more concerned with what is a real life scenario where we can use these Topologies. For example, I thought of maybe as clouds as an example of open sets of $\mathbb{R}^2$, since we cannot define exactly where these borders end. However maybe I am missing something important that makes everything to see much easier. – TeeJ Lockwood Dec 13 '15 at 04:57
  • Another example I thought of was cell phone reception/wifi reception. Perhaps another one could be explosions and their radius (standard topology). Perhaps another one could be how high a kite flies or simply the positioning of the kite (lower limit topology). I am curious about how storing information can be used in a topology. – TeeJ Lockwood Dec 13 '15 at 05:18
  • There is a tendency in texts to assume that a topology has to be defined in terms of open sets; this is an elegant definition but not the most intuitive; the one most related to the usual $\epsilon-\delta $ notions from analysis is the one in terms of neighbourhoods, see my book "Topology and Groupoids": http://groupoids.org.uk/topgpds.html . One then has to introduce open sets and show their relation to neighbourhoods, and the use of both in defining and using continuity of functions. – Ronnie Brown Jun 22 '16 at 11:54