Below I quote an answer that I posted. It was deleted. I do not know which users deleted it, so I cannot ask them what objections they had to it.
Does anyone know a reason why such a thing ought to be deleted?
If those who deleted it see this, can they tell me what they objected to? If I knew that, then I would know whether I agree with the objections, and if I found merit in them, I could have edited the posting before it was deleted, or I could take that information into account in future postings.
In some commonplace metric spaces such as $\ell^2,$ there are sets that are closed and bounded but NOT compact. In particular, the standard orthonormal basis of $\ell^2$ is an example of such a set. And the closed interval from $0$ to $1$ within the space of rational numbers with the usual metric is another example. These examples are closed and bounded but not compact.
Mathematicians have long been trapped within a general way of doing mathematics that tacitly encourages NOT motivating definitions. The definition of a "group", for example, is motivated by the fact that many concrete example have certain things in common. But textbooks say: "DEFINITION: A group is etc.etc.etc." It's dogma. Mathematics should be done non-dogmatically, and in some respects it is.
And here I will make a bolder claim: The pattern of thinking that results in this dogmatism among people who think their subject is done non-dogmatically is the SAME thing that causes required math courses in schools to in effect be taught dogmatically, by a mechanism that works in front of everybody's faces, except that they never look in that direction.