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I'm looking for a reference, if such a references exists.

So there are currently at least two approaches to topology.

  1. The point-set or "classical" approach to topology, which concerns itself with ordered pairs $(X,\tau)$ called topological spaces.

  2. The "pointless" approach to topology, which concerns itself with (particular kinds of) lattices $(\tau,\wedge,\vee)$ called frames. (For more information, see e.g. Wikipedia.)

I'm interested in a concept halfway between 1 and 2. We might call it "the classical approach, but with lattices."

In particular, rather than studying point-set topological spaces $(X,\tau)$, we concern ourselves with "lattice-theoretic" topological spaces $(P,\tau)$, 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 main motivation: We may be able to weaken the requirement that $P$ needs to be isomorphic to a powerset, and thereby obtain a more general theory, which is still classical in flavor.

Has this idea been studied before? If so, a reference recommendation would be great.

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Can you perhaps elaborate on how what you are looking for is different to locale theory? The only way I can interpret the above is that $\tau$ is a subset of P that is closed under arbitrary joins and finite meets. Weakening P from being a powerset lattice to just a complete lattice gives you locales precisely. So, unless you have some special subclass of lattices in mind, your question is not precise enough. – Ittay Weiss Jan 29 '13 at 9:08
@IttayWeiss You write, "The only way I can interpret the above is that $\tau$ is a subset of $P$ that is closed under arbitrary joins and finite meets." Yes this is precisely what I mean. This is different from a locale, which makes no mention of $P$ whatsoever. – goblin Jan 29 '13 at 9:18
ahhh, I see what you mean. Is the following the definition you have in mind: Suppose you have some functor F:Set-->CLat. Say that an F-topological space is a pair $(X,\tau)$ such that $X$ is a set and $\tau$ is a subset of $F(X)$, closed under arbitrary unions and finite meets. The case F(x)=powerset of X gives precisely ordinary topological spaces. – Ittay Weiss Jan 29 '13 at 9:25
To be honest, all that categorical stuff kind of defeats me... All I was thinking is that if $(X,\tau)$ is a topological space, then $\mathcal{P}X$ can be viewed as a lattice, so $(\mathcal{P}X,\tau)$ can be viewed as a "lattice-theoretic" topological space. Furthermore, we can forget about the fact that $\mathcal{P}X$ and $\tau$ are collections of sets. Who cares what they're collections of! What matters is how they relate. This is kind of like - who cares what the elements of a group are. Whether they're functions or matrices or whatever, doesn't matter. What matters is how they relate. – goblin Jan 29 '13 at 9:34
However, it's possible that what you're saying is more interesting than what I'm saying. – goblin Jan 29 '13 at 9:35
up vote 2 down vote accepted

I do not have a precise reference. But this may give you something to look at and/or some people to ask.

If you had asked me the similar question about convexity instead of topology, I would've given you a positive answer. The classical point-set topology satisfies the following axioms on the topology of closed sets $\tau \subseteq \mathcal{P}(X)$:

  1. $\tau \ni {\emptyset, X}$
  2. for any finite subset $K\subseteq\tau$ we have $\cup K\in \tau$
  3. for any subset $J \subseteq \tau$ we have $\cap J \in \tau$.

The collection of all convex subsets $C$ on an affine space $E$ satisfy a similar collection of properties:

  1. $C \ni {\emptyset, X}$
  2. for any subset $J \subseteq C$ we have $\cap J \in C$
  3. (optional, depending on definition) for any directed (with respect to inclusion) subset $S\subseteq C$ we have $\cup S \in C$

So one sees that there are certain similarities between convex structures and topological structures.

Now, for convex structures, something like what you proposed has been developed. It is sometimes called "abstract convexity", and one formulation is something like this:

Defn: Let $(X,\leq)$ be a complete lattice. A convexity system $\mathcal{C} \subseteq X$ is a subset that is closed under infimum. The system is said to be inductive if it is in addition closed under directed supremums.

Note that $X$ does not have to be the powerset lattice for a set. It turns out that many facts of convex analysis can be reproduced in this more general context. See, for example, Ivan Singer's Abstract Convex Analysis or MLJ van der Vel's Theory of Convex Structures.

Given the similarity between convex structures and topological ones (for example, the convex hull operator is almost a closure operator in the Kuratowski sense), maybe the sort of "topology as a subset of a complete lattice" point of view can be found linked to from the literature in abstract convexity, and maybe the experts in that area can point you in the right direction so-to-speak.

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Cool I'll check it out. I've actually encountered this concept before, except defined in the less general "point-set" fashion (rather than a "lattice-theoretic" fashion). It goes under the name "aligned space," in Coppel's Foundations of Convex Geometry. – goblin Jan 29 '13 at 13:04
Also, I think a few of your definitions are slightly off. Like 1 should feature a superset symbol, not an ownership symbol. – goblin Jan 29 '13 at 13:05
@user18921 corrected. I was changing between two sets of notations and evidently didn't make everything line up. Thanks. – Willie Wong Jan 29 '13 at 13:07
No worries. Also, and this is a technicality that EVERYONE get wrong, not just you, the collection of all closed sets in a topological space is not really closed under arbitrary intersections. In particular, the empty intersection gives the wrong answer - you wind up getting the universal class, rather than $X$. A little nicety of the lattice-theoretic approach is that this is automatically dealt with. – goblin Jan 29 '13 at 13:10
@user18921 that is more a matter of convention and brevity, which I am well aware of. For example in the more formal definition of the convexity system which I gave the membership requirement is conveniently omitted. – Willie Wong Jan 29 '13 at 13:26

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