What are the properties of the set of the Real Numbers without the Integers? This question came up in a lunchtime discussion with coworkers.  None of us are professional mathematicians or teachers of math.  I apologize for any incorrect math or sloppy terminology.
We were discussing getting from one number to another along the real number line.  The challenge is that we removed the integers from the number line.
So can an ant get from 0.5 to 1.5 by crawling along this "broken" line?
Searching the web, we discovered this may be a problem in topology, somehow (possibly) related to something called the "Long Line".  This is far outside of any of our knowledge so we would appreciate an explanation at roughly the basic calculus level.

The space you describe 
$...(-(n+1),-n)...(-2,-1)(-1,0)(0,1)(1,2)...(n,n+1)...$
is homeomorphic to countably infinite ("$\omega$") many copies of $(0,1)$. The Long Line you refer is made of uncountably infinite many copies of $[0,1)$. It can be thought of as a generalization of the nonnegative reals $[0,\infty)$, because we can view this space as  $\omega$-many copies of $[0,1)$:
$[0,1)[1,2)[2,3)...$
Your question about the ant is related to path connectedness (which is equivalent to connectedness here).  There is no continuous path from a point in $(n-1,n)$ to a point in $(n,n+1)$ as the subspace $(n-1,n)(n,n+1)$  is not connected, that is, it is the union of two nonempty open subsets. In the reals this would not be a problem as $(n-1,n+1)$ is connected.
