# “The operation is normal iff it's both monotone and continuous” — which math area studies operation?

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?

• 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. – Cameron Williams May 14 '14 at 4:01

• 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. – Dave L. Renfro May 14 '14 at 14:32