Since the time of Cantor, we have taught students to think of discrete sets as less than continuous intervals. Discrete sets of values have a property that they are countable, while continuous sets of values are not countable. We might think of this property as a limitation of discrete sets. The continuum has its own property. Continuous intervals have a property of a metric space. A metric space is equivalent to a topology with T2 / Hausdorff separability condition. The topology that is used so often that it known as the standard topology (especially regarding number systems) is the order topology. The order topology on the reals corresponds to an order on the reals, known as the standard order.

Even for a single real value, a topology is fixed by definition. There may be situations where the topology for a discrete set is not fixed. We can see the use of that topology in terms of an order. We can think of the continuum as having a unique order, while sets of discrete values do not have an implicit / unique order. There may be confusion with this property. Just because there is no unique order does not mean that an order cannot be selected. What we might think of as a natural order is the order provided by a generator (that matches a construction). There are infinitely many orderings for any countably infinite set. 

For simple examples of orderings it is easier to use discrete sets because of the requirements of the continuum. Students are likely to get the misleading impression that orders on the continuum are more complex than those on discrete sets. It should be emphasized that orders and topologies on discrete sets are less constrained.


If the continuum and discrete sets are incomparable, then a modern education should offer students the opportunity of alternative perspectives. For example, we can construct real numbers in terms of placeholder numbers. Then natural numbers can be formalized in terms of dimensions of real spaces or as extensions (square root of primes) of the real field. Another way to describe computation is in terms of an infinite sequence resulting from mapping a computation over a sequence of natural numerals. Countability could be described as the existence of a construction with a constant boundary. 


While I am on the subject of orders, I will mention cyclic orders. They are obscure, but they may deserve more attention. A cyclic order is to a linear order as the projective plane is to Euclidean space. They may be useful to orient otherwise unstructured data. There are plenty of uses for modular arithmetic too. The standard orders can be thought of as a specialization of cyclic orders (for infinite sets). In the case of finite sets, the topology does not match the cyclic order or we could say it matches more than one. The order topology on a cycle is undirected, so that it corresponds to an undirected cyclic order (i.e. an equivalence class where each order is paired with its reverse order). By considering graphs similar to topological spaces, we can see that we are also talking about Hamiltonian cycles.