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=== Topological properties of the real numbers ===
Many of the theorems of real analysis are consequences of the topological properties of the real number line. The order properties of the real numbers described above are closely related to these topological properties. As a [[topological space]], the real numbers has a ''standard topology'', which is the [[order topology]] induced by order <math><</math>. Alternatively, by defining the ''metric'' or ''distance function'' <math>d:\mathbb{R}\times\mathbb{R}\to\mathbb{R}_{\geq 0}</math> using the [[absolute value]] function as {{nowrap|<math>d(x, y) = |x - y|</math>,}} the real numbers become the prototypical example of a [[metric space]]. The topology induced by metric <math>d</math> turns out to be identical to the standard topology induced by order <math><</math>. Theorems like the [[intermediate value theorem]] that are essentially topological in nature can often be proved in the more general setting of metric or topological spaces rather than in <math>\mathbb{R}</math> only. Often, such proofs tend to be shorter or simpler compared to classical proofs that apply direct methods.