Let be a topological space and is a subset of , then the closure of is denoted by or is the intersection of all closed sets containing or all closed super set of . i.e. the smallest closed set containing .
On the other hand it can also be 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 closure (or adherence) of and is denoted by . In symbols
• Every set is always contained in its closure. i.e.
• Closure of a set by definition (being intersection of closed set is always 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 dense subset of . (i.e. dense in ), if
Consider the set of rational number (with usual topology), then the only closed set containing in . Which shows that . Hence is dense in .
• It may be noted that the set of irrational numbers is also dense in . i.e. .
• Rational are dense in and countable but irrational numbers are also dense in but not countable.