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DEFINITIONS Point of closure For ''S'' a subset of an Euclidean Space , ''x'' is a point of closure of ''S'' if every Open Ball centered at ''x'' contains a point of ''S''. (This point may be ''x'' itself.) This definition generalises to any subset ''S'' of a {''d''(''x'', ''s'') : ''s'' in ''S''} = 0. This definition generalises to Topological Space s by replacing "open ball" or "ball" with " Neighbourhood ". Let ''S'' be a subset of a topological space ''X''. Then ''x'' is a point of closure of ''S'' if every neighbourhood of ''x'' contains a point of ''S''. Note that this definition does not depend upon whether neighbourhoods are required to be open. Limit point The definition of a point of closure is closely related to the definition of a Limit Point . The difference between the two definitions is subtle but important — namely, in the definition of limit point, every neighbourhood of the point ''x'' in question must contain a point of the set ''other than x itself''. Thus, every limit point is a point of closure, but not every point of closure is a limit point. A point of closure which is not a limit point is an Isolated Point . In other words, a point ''x'' is an isolated point of ''S'' if it is an element of ''S'' and if there is a neighbourhood of ''x'' which contains no other points of ''S'' other than ''x'' itself. For a given set ''S'' and point ''x'', ''x'' is a point of closure of ''S'' If And Only If ''x'' is an element of ''S'' or ''x'' is a limit point of ''S''. Closure of a set The closure of a set ''S'' is the set of all points of closure of ''S''. The closure of ''S'' is denoted cl(''S''), Cl(''S''), or ''S''−. The closure of a set has the following properties.
Sometimes the second or third property above is taken as the ''definition'' of the topological closure. In a First-countable Space (such as a Metric Space ), cl(''S'') is the set of all Limits of all convergent Sequence s of points in ''S''. For a general topological space, this statement remains true if one replaces "sequence" by " Net ". Note that these properties are also satisfied if "closure", "intersection", "contains/containing", "smallest" and "closed" are replaced by "interior", "union", "contained in", "largest", and "open". For more on this matter, see Closure Operator below. EXAMPLES
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