According to the SEP:
Another place to find applications of inconsistency in analysis is topology, where one readily observes the practice of cutting and pasting spaces being described as “identification” of one boundary with another.
For example, joining the segments [-1,0] & [0,1] at 0.
One can show that this can be described in an inconsistent theory in which the two boundaries are both identical and not identical, and it can be further argued that this is the most natural description of the practice.
Now points do not have boundaries, so we can't construct a line by 'joining' points, as one might imagine (the usual construction is to give (not join) a bare set of points and then add the right topology); but can we consider a limiting argument? That is:
Consider, in one picture, that the real line is made of line segments, say of unit size; and joined end-to-end; as in the example above; using inconsistent joins.
Then take the limit as the interval size goes to zero.
Does this work as a definition of the inconsistent real line?