Let be a topological space and be a subset of , then the closure of is denoted by or is the intersection of all closed sets containing or all closed super sets of ; i.e. the smallest closed set containing .
On the other hand it can also be written as let be a topological space and let be any subset of . A point is said to be adherent to if each neighborhood of contains a point of (which may be itself). The set of all points of adherent to is called the closure (or adherence) of and is denoted by . In symbols:
• Every set is always contained in its closure, i.e.
• The closure of a set by definition (the intersection of a closed set is always a closed set).
Let with topology and be a subset of .
Open sets are
Closed sets are
Closed sets containing A are
Theorem: Let be a topological space, and and be subsets of , then
• is closed if and only if
Dense Subset of a Topological Space
Let be a topological space and be a subset of , then is said to be a dense subset of (i.e. dense in ), if
Consider the set of rational numbers (with usual topology), then the only closed set containing in . This shows that . Hence is dense in .
• It may be noted that the set of irrational numbers is also dense in , i.e. .
• Rational numbers are dense in and countable but irrational numbers are also dense in but not countable.