# Are questions of the form "has this ever been studied?" appropriate?

Are requests of the form "has this ever been studied, and if so, may I please have a reference?" considered appropriate?

But let me be more specific.

So firstly, there's the point-set or "classical" approach to topology, which concerns itself with ordered pairs $(X,\tau)$ called topological spaces.

Then there's the pointless approach to topology, which concerns itself with lattices $(\tau,\wedge,\vee)$ called frames (in which finite meets distribute over arbitrary joins.)

I'm interested in a concept halfway between the two. We might call it "the classical approach, but with lattices." Rather than $(X,\tau)$, we concern ourselves $(P,\tau),$ where $P$ is a lattice that is isomorphic to a powerset lattice, and $\tau$ is a subset of $P$ that is closed with respect to arbitrary joins etc.

The motivation for this idea is as follows: we may be able to weaken the requirement that $P$ needs to be isomorphic to a powerset, and still be able to develop classical topology just fine.

So my question is, has this idea been studied before, and if so, may I please have reference recommended?

• One problem with this type of question is that they are semi-answerable: if the object has been studied, one can point to a reference, but there is no way to show that the subject has not been studied. It may have been studied, for example, and found to be useless so nothing was written as a result. Jan 28 '13 at 17:41
• @Mariano This comes to mind... ;-) Jan 28 '13 at 20:07
• @MichaelGreinecker, well, that is somewhat different. There can't be a standard notion of something for which one cannot give references! :-) Jan 28 '13 at 20:38
• Ask away, but now, ask it at math.stackexchange, the main site! Feb 1 '13 at 18:05