Density and continuua for sets without linear orders

Question

A dense order is defined as an ordered set such that for any $x$ and $y$ such that $x < y$, there is a $z$ such that $x < z < y$.

A linear continuum is defined as a linearly ordered set that is both dense and complete (i.e. has the least upper bound property).

Is there a natural way of generalizing these definitions to sets that aren't linearly ordered? In particular, I'm thinking about sets with a metric such that (intuitively) between any two elements there is another element (for density) or a continuum of other elements (for continuity).

Thank you.

Answer 1

If you think of density as "between any two elements there is a third element", then density doesn't have a common direct analogue in general metric spaces. One alternative for a metric space $X$ with distance function $d$ is

For any $x\neq y\in X$, there is a $z\in X$ with $z\neq x,y$ such that $d(x,z)+d(z,y)=d(x,y)$

This says that $X$ has some point "directly between" any two points. This is a rather strict requirement, and not always satisfied in spaces you might want it to be. For instance, the unit circle in the plane with the metric defined as the length of the straight line segment between two points doesn't fulfill this property.

A weaker alternative is

For any $x\neq y\in X$, there is a $z\in X$ with $z\neq x,y$ such that $d(x,z), d(z,y)<d(x,y)$

This says that if you want to go from $x$ to $y$, but the distance is too big to go in a single step, there is an intermediate point $z$ that makes each of the two steps shorter. To borrow some geometric intuition, it roughly says that any line segment is the longest side in some triangle.

On the other hand, if you think of density as "there are points in the space which are arbitrarily close to any given point", then this does indeed translate directly to metric spaces as the property

For any $x\in X$ and any $\varepsilon>0$, there is a $y\in X$ with $y\neq x$ such that $d(x,y)<\varepsilon$

This is called "having no isolated points". It is also closer to what the word "dense" is used for in real analysis.

The least upper bound property has a very standard analogue. It's called a complete metric space. We say that a metric space $X$ is complete iff the following holds:

Any Cauchy sequence in $X$ converges

The notion of "Cauchy" captures the essence of "monotone and bounded above" for when ordering isn't an option. More specifically, a sequence $x_n\in X$ is Cauchy iff

For any $\varepsilon>0$, there is an $N\in \Bbb N$ such that $d(x_m, x_n)<\varepsilon$ whenever $m,n\geq N$

Answer 2

There are a couple of generalisations that may be of interest to you. In order theory, the notion of completeness makes sense for orders that are not linear, and simply means that every set of elements has both a greatest lower bound and least upper bound.

In metric spaces, the `equivalent' notion of density here would probably be that the space was dense-in-itself. In other words, every point $x$ of your metric space $M$ should be the limit of some sequence of points in $M-\{x\}$.

Metric spaces do have a notion of completeness. This is that every Cauchy sequence in the space converges to a limit point in the space. However, if you want a continuum of elements that "lie between" any other two, you will need more than this, since your metric space may be in two far apart pieces. You probably want what is called a "length space", which is a metric space where the distance between two points is the greatest lower bound of the lengths of all paths between two points. Here a short path between two points $x$ and $y$ gives you a continuum of points approximately between $x$ and $y$.

Answer 3

There is a subclass of topological spaces called LOTS (linearly ordered topological spaces), which is a linearly ordered set $(X,<)$ with as its topology the order topology (with as subbase the standard upper and lower sets).

It turns out (and Munkres essentially shows this in his text, and it's a classic fact) that $X$ as a topological space is connected iff it is order dense and order complete. (So the order is a linear continuum, in Munkres' terminology). This is a nice illustration of how a purely topological fact about the space can be characterised in pure order terms, because of the order topology connection. Many such characterisations and facts about LOTS are known (it was a popular subject in topology in the 70's and 80's); they form quite a special class. There are generalisations (GO-spaces) and special metrisation theorems that apply to LOTS and theorems that allow us to determine whether a given topological space is actually a LOTS (even when the order might not be pre-given), like metrisation theorems are to metric spaces.

Long story short: the "generalisation" of a linear continuum is just a connected space (i.e. the corresponding notion from LOTS-es vis à vis general spaces). Historically it's the other way around of course: we knew connected spaces and when you look for what a connected LOTS is, you get exactly the linear continua.