I just read Enderton's "Elements of Set Theory" to have a basic understanding of sets (btw it's a great book). One line of it says: "the operation is normal iff it's both monotone and continuous."

I'm wondering, the terms, normal, monotone, continuous, seems not come from the thin air. are these some common terms in a specific math area that studies operation/operator ? is it algebra?

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    $\begingroup$ This can go one of two ways. If you mean abstract operations on algebraic sets, then I think the area would be along the lines of universal algebras. If you mean operators on functions and vectors and such, it would be (functional) analysis. $\endgroup$ – Cameron Williams May 14 '14 at 4:01

The part of Enderton you quote is his own definition of the word “normal” for a particular thing he calls an “operation.” “Normal” is perhaps the most overused word in all of mathematics, but I'm not a set theorist, so I can’t say where his use of it was inspired. It appears in every subfield of mathematics with widely different meanings.

Enderton’s “operations” are basically functions. He doesn’t call them functions only because their domain (all ordinals) is not a set. The word “monotone” comes from analysis mostly, and “continuous” comes from analysis or topology.

All fields of mathematics overlap, but if you want to find out more about monotone, continuous things, a book on analysis, and perhaps a gentle one on topology as well, would be good places to start.

  • $\begingroup$ I agree about the overuse of "normal", but this usage of "normal" by Enderton is standard in set theory (since at least the 1960s). A google search for "normal function" {AND} "ordinal" {AND} "transfinite" will show this. For certain special cases the results are due to Cantor (e.g. the function ${\omega}^{x}$ is continuous and increasing, so it has fixed points, which Cantor called $\epsilon$-numbers) and the general formulation is due to Veblen (his 1908 paper Continuous increasing functions of finite and transfinite ordinals. $\endgroup$ – Dave L. Renfro May 14 '14 at 14:32

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