Hello. I have several remarks about your work, first about general topology, then about physics.
I also considered this problem, of how else topology might be formalized so as to be more convenient for different purposes, may it be easiness of use, restrictions or generalizations, or modified kinds of structures (usually more rigid than some topology, since topology is usually the lightest non-trivial interesting structure that may be found in a given space), and I came to consider other options, I will share with you.
First, I agree with you about the importance of linear orders: it is a remarkable property of lines that their topology is equivalently expressible in that very simple formalism (linear order), unlike other kinds of topological spaces, so we may consider to use this fact to define general topological spaces by somehow "reducing" them to this 1-dimensional case, explicitly involving lines somewhere in the general definition of topology. However there are different possible ways of doing so, and we need to see the larger picture of possible methods and how they may differ, in order to find out which method may be more relevant for given purposes, instead of picking up a fixed method just because it is the first method we happen to think of.
Let us consider this problem in its abstract generality : if there is a general kind of "spaces" we have intuitive ideas of, but we do not know how to formalize (by which kind of operation or relation, as none seems to fit), then can we still find a method to "anyway simply formalize it no matter what a subtle kind of system it is", that will be general enough to include topology ?
As surprising as it may seem, the answer is yes.
Let us explain this amazingly general and still simple toolbox.
First step is to look at the intended kind of "spaces" as forming a category: no matter how topology may be formalized, we expect the concept of "continuous function" between spaces to make sense. We have a clear intuition what it means for a function to be continuous, so there should exist a definite set of all continuous functions from a space to another, the only problem is that we cannot write a definition for it since we did not formalize what is the topological structure we want these functions to preserve. Never mind, we will define this later. At first, let us just admit this concept of continuous function as primitive, forming a category.
Now the problem is: starting with an arbitrary category, is there any general, systematic way to interpret its objects as systems with a kind of structures, so that the morphisms in this category will happen to preserve these structures, and that will still be interesting even if (as in usual topological spaces) there does not exist any non-trivial invariant algebraic operation in these spaces ? We can make it, and here is how.
First we need to pick up a fixed object K in the category, with the quality of being "flexible enough" to serve as the prototype from which the structure of other objects will be formalized. For the needs of expressing topology in physics, we can take K=R (the line of real numbers).
For even more general cases where one fixed object (space) is not "flexible enough", we can take a series of different spaces that would hopefully "complete each other" in their exhaustion of the different kinds of possible shapes.
Now with a fixed K, here is how to interpret any object M in the category as a system with structures. In fact there are 2 structures we can automatically define on all objects M, that will be preserved by all morphisms in the category:
1) The set Mor(K,M) of all morphisms from K to M;
2) The set Mor(M,K) of all morphisms from M to K.
Indeed, for any objects M and N, any morphism from M to N maps Mor(K,M) into Mor(K,N), and also Mor(N,K) into Mor(M,K). I wrote a page with a representation theorem that goes deeper than what I found usually done elsewhere on the topic.
Now as compared to this, your approach can be described as follows:
- You take as K not just the real line, but all possible linearly ordered sets;
- You only consider the embeddings of such K into all possible M, instead of any continuous functions. You refuse those mapping 2 non-neighbor points into neighbor points (in the case of finite graphs).
Some questions:
- It may look good to handle continuous lines as well as discrete lines in similar ways, however what is the use of "mixing" them by not formally assuming K as fixed, so as to admit the case of hybrid systems ? I imagine that sometimes we need to formalize continuous spaces, other times discrete graphs, but I hardly see the interest of hybrid systems. Anyway, there is no interesting continuous map in either way between a discrete and a continuous line.
- There is a diversity of non-isomorphic linear orders, beyond finite ones and that of R. Your formalism automatically admits them. Are you interested in such generalizations, despite the fact that only the case of R comes in usual theories of physics, namely in general relativity which seems to be what you focus on ? Examples : the set of rationals ; the set of irrationals ; the long line ; lines where all countable monotone sequences converge but that are incomplete since gaps have uncountable cofinality; Suslin lines, whose existence is undecidable in ZF.
- By only accepting embeddings of lines into your topological spaces, you make it straightforward to define embeddings between them, however it leaves the concept of other continuous functions less straightforward to define. Do you see the notion of embedding as a more important correspondence between topological spaces than continuous functions ?
I will write more comments later.
