Let  be a topological space and A is a subset of X, then the closure of A is denoted by or  is the intersection of all closed sets containing A or all closed super set of A. i.e. the smallest closed set containing A.
            On the other hand it can also be as let be a topological space and let A be any subset of X. A point is said to be adherent to A if each neighborhood of contains a point of A (which may be itself). The set of all points of X adherent to A is called closure (or adherence) of A and is denoted by. In symbols
                       

Remarks:

• 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).

Example:
            Let  with topology  and  be a subset of X.
            Open sets are
            Closed sets are
            Closed sets containing A are
            Now

Theorem: Let be a topological space, and A and B be subsets of X, then

• A is closed if and only if
• 
• 
• 
• 

Dense Subset of a Topological Space:
            Let be a topological space and A be a subset of X, then A is said to be dense subset of X. (i.e. dense in X), if

Example: Consider the set of rational number  (with usual topology), then the only closed set containing in. Which shows that . Hence  is dense in.

Remarks:

• 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.