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I'm trying to build an intuition/understanding of the notion of a topology.

I've read the following thread on MO https://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets/19691#19691 and several others here, like: Intuition behind topological spaces and still could not convince myself that I've understood the notion properly.

Things I think I've understood are:

When we go 'top-down' and strip spaces off the angles, distances, norms, etc. What we left with is a Topological Space of a form (X, T), which deals with a ground set X and some other explicitly chosen set T that we call Topology and which serves as a structure definition for X. It seems that, since there is no "distance" anymore it is left to explicitly define the relations between the elements of X, which is now the job of T. A case of topologies induced by a metric is rather a special case and not the general.

Q: I am interested in the general case of topologies, not induced by a metric. If say, I set T explicitly for my X like this: $$X = \{A,B,C,D,E\}$$ $$ T = \{X, \emptyset, \{A, B\},\{B, C\}, \{A,B,C\},\{B\}\}$$

Does T here somehow define, which elements of X are near to / far from every other element? Since there is no distance here defined, then my guess is that "far/near" will be a binary value, which will tell us which elements from X are connected to each other and which are not, making X look like a non-directed graph.

Also, how exactly do we choose the rule by which we say that, e.g. A is near/connected to B or not? Is it fixed or freely defined?

Furthermore, if above is the case, then isn't this "rule" also something like a metric?

Could you please list these connections in the above X according to T? If I am not wrong, then they are all unconnected / far from each other.

Eric Wofsey
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    Well, when you drop the metric, you lose the notion of nearness. We can make the topology into a poset and that will measure where things converge to, however topology is not meant to deal with "nearness", it is more about convergence or continuity of stuff. You want to compare a lot of points to each other, and not just make a statement about two points being close. – Severin Schraven Dec 26 '19 at 17:15
  • Then what exactly T represents / shows in my example above? What does it tell in this specific case about X? If I defines a different topology, say T1, how would it change the way we look at X? –  Dec 26 '19 at 17:16
  • You may be interested in Kolmogorov spaces. Also, a Topology is more about sets rather than points. You could also regard a Topology as a stronger form of a Graph. – Somos Dec 26 '19 at 17:21
  • Of possible interest might be parts of this answer, especially the paragraph near the beginning that starts with "Roughly speaking", the next paragraph, and the paragraph further down that starts with "One the things that (b) in Theorem 4". Also, for one way to arrive at a topological space by specializing from more generalized notions, in contrast to the usual situation in which one arrives at a topological space by generalizing from more specialized notions, see my 3-part answer to this question. – Dave L. Renfro Dec 26 '19 at 18:13

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