In order theory, is there a specific name for functions that preserve the limits of increasing/decreasing sequences?
My best find was Scott-continuity, but it's a generalization for directed sets. The definitions I've see for preserving infima/suprema don't cut it either. I think an equivalent form might be "preserves the suprema of countable chains", but I didn't give that much thought and I still need the name for it.
Right now I'm left with saying "f is right-side sequentially continuous" (RSSC). Idk if the "right" part even applies for any ordered set in general.
Examples:
- In real analysis RSSC is equivalent to the usual right-side continuity.
- Meassures are continuous in this sense: In every monotonic sequence of sets (given by inclusion) the limit is the union/intersection of them. The sigma-additivity of meassures mean that the meassure of the limit is the limit of the meassures.
- Let f be an ordinal mapping that is the identity almost everywhere, but maps Alpeh_1 to (Alpeh_1 + 1). This mapping is LSSC (I think trivially RSSC too) but not continuous.
- The mapping S : R --> Poset(R) given by S(x) = (-inf, x] is RSSC but not LSSC.
The motivation for this question is doing this more elegant proof in probability:
A cumulative distribution function F is always right-side continuous because:
- In the real numbers it can be characterized as RSSC.
- F is the composition of a probability meassure with the function "S" defined above.
- The composition of two RSSC functions is also RSSC.