# Closure of a Set

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

**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 .

Open sets are

Closed sets are

Closed sets containing A are

Now

**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

**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.