# Closure of a Set

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:

**Remarks:**

• 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).

**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 a dense subset of (i.e. dense in ), if

**Example:**

Consider the set of rational numbers (with usual topology), then the only closed set containing in . This shows that . Hence is dense in .

**Remarks:**

• 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.