Idea behind definitions in math

Question

Maybe this is not such a great question for this site as it is a bit open ended, but I will ask it anyways. I was wondering if anyone has any insight into the motivation for quote unquote "major" definitions in mathematics. Take, for example, the notion of compactness in a metric space. I was taught that a set $A \subset M$ is compact if every open cover of $A$ has a finite subcover (along with equivalent definitions, but let's focus on this one). This is good and all, but who came up with this definition and why is it useful? It seems to me that the rigor of this definition is a bit arbitrary. Was the definition useful in proving some groundbreaking theorem? Why are compact sets defined to begin with?

Answer 1

For the specific case of compactness, see this paper:

A pedagogical history of compactness, by Manya Raman-Sundstrom, arXiv:1006.4131

"This paper traces the history of compactness from the original motivating questions, through the development of the definition, to a generalization of sequential compactness in terms of nets and filters."

Answer 2

As for your open-ended question, definitions almost always appear as an outgrowth of discoveries. For example, one mathematician obtains an important result; other mathematicians then comb over the proof, trying to isolate "what's really going on", and distill a key concept. Variations on this paradigm occur, but as a rule the results come first, the definitions later.

Compactness is a good example, and the paper cited in the other answer traces the history. Here's another one: the definition of a group.

The story starts, more or less, with the discovery in 16th C. Italy of the solutions to the cubic and quartic equations. As you probably know, attempts to solve the quintic proved fruitless. In the 18th C., Lagrange writes a paper in which he attempts to analyze what was really going on with the cubic and quartic formulas, and why the approach didn't work for n=5. He isolates permutations of the roots as the key concept.

Building on Lagrange's insight, Ruffini, Abel, and Galois make dramatic progress in the theory of equations. People are now talking about groups of permutations, but not the abstract concept.

In the 19th C., Cayley takes this background and uses it (together with other threads, like Gauss's work in number theory) to define the abstract concept of a group.

Another example, more recent: in the 19th C., combinatorial topology gets started, thanks to work by Euler, Listing, Poincare, and others. Out of this work arises the Betti numbers of a manifold. In modern terms, the Betti numbers are the ranks of the homology groups. In the 20th C., Emmy Noether shifts the emphasis from the numerical invariants (the Betti numbers) to the groups. (I'm not sure if Noether was the first to explicitly define the homology groups.)

In the second half of the 20th C., as different homology and cohomology theories proliferate, Eilenberg and Steenrod write a book, trying to unify what's been done in a common axiomatic framework. Soon thereafter, Eilenberg and MacLane write their book "Homological Algebra". The general definition of a category results from all this development.

Of course, the new definitions lead to new results, a never-ending cycle. As quoted in the compactness paper:

Modern mathematics tends to obliterate history: each new school rewrites the foundations of its subject in its own language, which makes for fine logic but poor pedagogy. --R. Hartshorne