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.

From The Continuous and the Infinitesimal by John L Bell, p280: 'Taken together, these principles have been shown to have remarkable consequences for the indecomposability of subsets of the continuum. Not only is the intuitionistic continuum indecomposable, but, assuming the Continuity Principle and Kripke's Scheme it remains indecomposable even if one pricks it with a pin. [More exactly, for any real number a, the complement R - {a} of {a} is indecomposable.] "The [intuitionistic] continuum has, as it were, a syrupy nature, one cannot simply take away one point." [In Van Dalen (1997) the classical continuum is described as the "frozen intuitionistic continuum".] If in addition Bar Induction is assumed, then, still more surprisingly, indecomposability is maintained even when all the rational points are removed from the continuum.'  
