What is a topological space good for?

I know there are already some questions similar to this, which all give an answer that a topological space creates some structure on a set which is an abstraction of distance and makes it possible to define other concepts like connectedness, compactness, metrizability asf. I understand this in theory, but when I create a simple example I fail to grasp what distance/closeness means in a topological space.

For example, when I have the following set X and a set of subsets τ1 which does not satisfy the axioms of a topological space:
X={a,b,c},τ1={X,,{a,b},{b,c}}

now I make this into a topological space by adding the sets {c} and {b}

X={a,b,c},τ2={X,,{a,b},{c},{b,c},{b}}

What is “better” now and how is this related to some notion of distance or closeness?

In other words, what did I gain by adding the subsets {c} and {b} to τ2?

More over what can I do with τ2 I can’t do with τ1?

Answer

I like to think of topological spaces as defining “semidecidable properties”. Let me explain.

Imagine I have an object that I think weighs about one kilogram. Suppose that, as a matter of fact, the object weighs less than one kilogram. Then I can, using a sufficiently accurate scale, determine that the object weighs less than one kilogram. Even if the object weighs, say, 0.9999996 kilograms, all I need to do is find a scale that’s accurate to within, say, 0.0000002 kilograms, and that scale will be able to tell me that the object weighs less than one kilogram.

This means that “weighing less than one kilogram” is a semidecidable property: if an object has the property, then I can determine that it has the property.

Suppose, on the other hand, that the object actually weighs exactly one kilogram. There’s no way I can measure the object and determine that it weighs exactly one kilogram, because no matter how precisely I measure it, it’s still possible that there’s some amount of error which I haven’t discovered yet. So “weighing exactly one kilogram” is not a semidecidable property.

What does this have to do with topological spaces? Well, an open set in a topological space corresponds to a semidecidable property of that space. This is why in the topological space of real numbers, the set {x:xR,x<1} is an open set, but the set {x:xR,x=1} is not.

So, consider the "topological space" X={a,b,c} with open sets , {a,b}, {b,c}, and {a,b,c}. In this "topological space", you are asserting that

  • (since {a,b} is open) if you have a point which is either a or b, then it is possible to measure it and determine that it is either a or b (though it is not necessarily possible to determine which one it is);
  • (since {b,c} is open) if you have a point which is either b or c, then it is possible to determine that it is either b or c; but
  • (since {b} is not open) if you have the point b, it is not possible to determine that it is b.

However, these assertions contradict each other. Suppose that you have the point b. Because of the first bullet point, there is some measurement you can make which will tell you that the point is either a or b. And because of the second bullet point, there is another measurement you can make which will tell you that the point is either b or c. If you simply make both of these two measurements, then you will have successfully determined that the point is (either a or b, and either b or c)—in other words, that the point is b. But the third bullet point asserts that this is impossible!

For more explanation of this idea, see these two answers:

Attribution
Source : Link , Question Author : holistic , Answer Author :
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