balls of radius that are centered at a point in ): It directly follows that an -neighbourhood is a uniform neighbourhood, and that a set is a uniform neighbourhood if and only if it contains an -neighbourhood for some value of Examples Given the set of real numbers with the usual Euclidean metric and a subset defined as then is a neighbourhood for the set of natural numbers, but is a uniform neighbourhood of this set. Topology from neighbourhoods The above definition is useful if the notion of open set is already defined. There is an alternative way to define a topology, by first defining the neighbourhood system, and then open sets as those sets containing a neighbourhood of each of their points. A neighbourhood system on is the assignment of a filter of subsets of to each in such that the point is an element of each in each in contains some in such that for each in is in One can show that both definitions are compatible, that is, the topology obtained from the neighbourhood system defined using open sets is the original one, and vice versa when starting out from a neighbourhood system. Uniform neighbourhoods In a uniform space is called a uniform neighbourhood of if there exists an entourage such that contains all points of that are -close to some point of that is, for all Deleted neighbourhood A deleted neighbourhood of a point (sometimes called a punctured neighbourhood) is a neighbourhood of without For instance, the interval is a neighbourhood of in the real line, so the set is a deleted neighbourhood of A deleted neighbourhood of a given point is not in fact a neighbourhood of the point. The concept of deleted neighbourhood occurs in the definition of the limit of a function and in the definition of limit points (among other things). See also References General topology Mathematical analysis