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As long as we're doing this, show why it is important in topology.
In the field of [[topology]], an important difference between the rational and real numbers is that the rational numbers are "totally disconnected"<!--No wikilink; I doubt that it would ever change color from red.-->, whereas the real numbers are [[connected (topology)|connected]]. To see this, observe that the set of numbers strictly less than <math>\sqrt{2}</math> is open in both sets, but it is also closed in the rationals but not in the reals. In each case, it has a [[limit point]], namely, <math>\sqrt{2}</math>, which it clearly does not contain. In the reals, the fact that <math>\sqrt{2}</math> is not contained makes the set not closed, whereas in the rationals, it doesn't matter because that point doesn't exist. In the rationals, the fact that the set is both open and closed makes it disconnected from its complement. Any open set in the rationals contains situations like this, which makes the rationals totally disconnected.
In the field of [[topology]], an important difference between the rational and real numbers is that the rational numbers are "totally disconnected"<!--No wikilink; I doubt that it would ever change color from red.-->, whereas the real numbers are [[connected (topology)|connected]]. To see this, observe that the set of numbers strictly less than <math>\sqrt{2}</math> is open in both sets, but it is also closed in the rationals but not in the reals. In each case, it has a [[limit point]], namely, <math>\sqrt{2}</math>, which it clearly does not contain. In the reals, the fact that <math>\sqrt{2}</math> is not contained makes the set not closed, whereas in the rationals, it doesn't matter because that point doesn't exist. In the rationals, the fact that the set is both open and closed makes it disconnected from its complement. Any open set in the rationals contains situations like this, which makes the rationals totally disconnected.
The fact that the reals are topologically connected makes them the standard starting point for many topological topics, such as [[homotopy]], [[homology]], and [[manifold]]s.
==History==
==History==