Editing Open set (section)

== Motivation ==
Intuitively, an open set provides a method to distinguish two [[Point (geometry)|points]]. For example, if about one of two points in a [[topological space]], there exists an open set not containing the other (distinct) point, the two points are referred to as [[topologically distinguishable]]. In this manner, one may speak of whether two points, or more generally two [[subset]]s, of a topological space are "near" without concretely defining a [[Metric (mathematics)|distance]]. Therefore, topological spaces may be seen as a generalization of spaces equipped with a notion of distance, which are called [[metric space]]s.

In the set of all [[real number]]s, one has the natural [[Euclidean metric]]; that is, a function which measures the distance between two real numbers: {{math|1=''d''(''x'', ''y'') = {{mabs|''x'' − ''y''}}}}. Therefore, given a real number ''x'', one can speak of the set of all points close to that real number; that is, within ''ε'' of ''x''. In essence, points within ε of ''x'' approximate ''x'' to an accuracy of degree ''ε''. Note that ''ε'' > 0 always but as ''ε'' becomes smaller and smaller, one obtains points that approximate ''x'' to a higher and higher degree of accuracy. For example, if ''x'' = 0 and ''ε'' = 1, the points within ''ε'' of ''x'' are precisely the points of the [[Interval (mathematics)#Notations for intervals|interval]] (−1, 1); that is, the set of all real numbers between −1 and 1. However, with ''ε'' = 0.5, the points within ''ε'' of ''x'' are precisely the points of (−0.5, 0.5). Clearly, these points approximate ''x'' to a greater degree of accuracy than when ''ε'' = 1.

The previous discussion shows, for the case ''x'' = 0, that one may approximate ''x'' to higher and higher degrees of accuracy by defining ''ε'' to be smaller and smaller. In particular, sets of the form (−''ε'', ''ε'') give us a lot of information about points close to ''x'' = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to ''x''. This innovative idea has far-reaching consequences; in particular, by defining different collections of sets containing 0 (distinct from the sets (−''ε'', ''ε'')), one may find different results regarding the distance between 0 and other real numbers. For example, if we were to define '''R''' as the only such set for "measuring distance", all points are close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member of '''R'''. Thus, we find that in some sense, every real number is distance 0 away from 0. It may help in this case to think of the measure as being a binary condition: all things in '''R''' are equally close to 0, while any item that is not in '''R''' is not close to 0.

In general, one refers to the family of sets containing 0, used to approximate 0, as a '''''neighborhood basis'''''; a member of this neighborhood basis is referred to as an open set. In fact, one may generalize these notions to an arbitrary set (''X''); rather than just the real numbers. In this case, given a point (''x'') of that set, one may define a collection of sets "around" (that is, containing) ''x'', used to approximate ''x''. Of course, this collection would have to satisfy certain properties (known as '''axioms''') for otherwise we may not have a well-defined method to measure distance. For example, every point in ''X'' should approximate ''x'' to ''some'' degree of accuracy. Thus ''X'' should be in this family. Once we begin to define "smaller" sets containing ''x'', we tend to approximate ''x'' to a greater degree of accuracy. Bearing this in mind, one may define the remaining axioms that the family of sets about ''x'' is required to satisfy.