Integers have definable “immediate neighbours”; reals do not. Is this a consequence of density? Every integer has an immediate neighbour to the right.  By this, I mean that given an integer $z$, one can define a neighbourhood to the right of $z$ of length $R$ that does not contain $z$ (essentially half of a deleted neighbourhood). Call it $$N_+^*(z,R)=\{z+r:0<r<R\}$$ There exists at least one $R\in\mathbb R^+$ such that $N_+^*(z,R)$ contains only one element.  This is the neighbour of $z$ to the right. For example, we could say that $1$ is the neighbour of $0$ to the right because $N_+^*(0,1.5)$ contains only the element $1$.

The real numbers on the other hand do not have immediate neighbours.  This is because for any real number $r$ and any supposed immediate neighbour $n$, there exists an even closer neighbour, $(n+r)/2$.
This looks a lot like the definition of a limit point.  This leads me to believe that integers have immediate neighbours because $\Bbb Z$ is not dense in $\Bbb R$, and reals do not have immediate neighbours because $\Bbb R$ is dense in $\Bbb R$. However I’m no topologist.  Is my intuition correct?
 A: As requested by the asker in comments I am expanding my comment into an answer.
The concept of neighbor/neighborhoods is essentially dependent on order relation. And hence it is best to have an idea of how the order relations work in systems like $\mathbb{Z}, \mathbb {Q}, \mathbb{R} $.
The order relation is defined in these systems by first identifying a set $P$ of positive numbers with the following properties:


*

*A number in system is either $0$ or belongs to $P$ or its negative belongs to $P$.

*The sum and product of two members of $P$ are also members of $P$.


And once we have identified this set of positive numbers we define $x>y$ if $x-y$ is a positive number. And we have $x<y$ if $y>x$. 
Thus given a number we can find a greater one by adding a positive number to it (and a smaller one by subtracting a positive number from it).
For defining neighbor we have to bring in the metric / distance in picture. If $a, b$ are two numbers then $|a-b|$ is said to be the distnace between them. A neighbor of a number $x$ is supposed to another number $y$ such that the distance between $x$ and $y$ is small (this matches the idea of neighbor in real life). In $\mathbb{Z} $ we have a least positive number $1$ and hence given a number $x\in\mathbb {Z} $ there are two nearest neighbors to $x$ namely $x+1$ and $x-1$. Any other integer lies at a greater distance from $x$ compared to these two numbers. The typical terms for these numbers are successor and predecessor of $x$.
In $\mathbb{Q} $ there is no smallest positive number because given a positive rational $m/n$ we have a smaller one $m/(n+1)$. This is clearly based on the fact that given a positive integer $n$ there is a greater one namely $n+1$ (thus there is no largest positive integer). Due to this we don't have a concept of neighbor in $\mathbb {Q} $. Given any number $x\in\mathbb{Q} $ we can find rationals $y, z$ such that $y<x<z$ and the distances of $y$ and $z$ from $x$ are as small as we please.
Then we circumvent the problem here by defining a neighborhood of $x$. We take two such numbers $y, z$ with $y<x<z$ and all numbers between $y$ and $z$ are said to form a neighborhood of $x$.
The system $\mathbb{R} $ of real numbers is exactly similar to the system $\mathbb{Q} $ (except for the completeness property). When the development of $\mathbb {R} $ is done based on $\mathbb {Q} $ (eg via Dedekind cuts or Cauchy sequences) then all the properties of $\mathbb{Q} $ are preserved and we get an extra thing called completeness precisely because of the development done.
Thus there is no smallest positive real number and the notion of neighborhood is developed exactly like that in the system of rationals.
The notion of density (related concepts like limit points etc) are also based on the notions described above. Thus integers are not dense because the smallest distance between two distinct integers is $1$. The integers thus act as isolated points.
The whole abstraction of metric spaces is also based on the same idea that there is no smallest positive real number. Later some people got fed up of giving so much importance to reals and they invented an abstraction of topological spaces where reals don't matter and neighborhood are defined almost arbitrarily. 
As a final remark, easier parts of real analysis are built on the fact that there is no smallest positive real number and difficult parts are based completeness of reals. 
