[eli5] What is topological equivalence and why is it significant?
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I’ll try to do this in a few steps.
In math, finding equivalences between different ideas or objects is considered “interesting”. This is because these equivalences and their nature helps us make statements about properties of these different objects. Analogy: Samsungs are equivalent to iPhones in terms of their ability to make phone calls, text, and browse the web. So if you need to borrow your friend’s phone to text your boss, it doesn’t matter to you if he has a Samsung or an iPhone.
Objects are equivalent in a topological sense if there’s “continuous map” between them. This is just another way to say “all relevant topological information is the same between the objects”, even if the objects are not exactly the same outside topology. An often cited example is that of a coffee mug and a donut with a hole. We all know a coffee mug behaves very differently from a donut in real life, and even just the shapes are recognizably different. However from a topological sense they may as well be the same – they both have one hole, which is the important thing that matters. As a counter example, a ball is not equivalent in this sense to a donut because they have different number of holes.
Continuing the analogy from above, if you needed a donut for some topological use (like using it to make a math proof), you may borrow your friend’s coffee mug and be satisfied, but you may not be satisfied with your other friend’s ball.
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