## 5.1. Open and Closed Sets

### Definition 5.1.5: Boundary, Accumulation, Interior, and Isolated Points

Let

**S**be an arbitrary set in the real line**R**.- A point
*b*is called**R****boundary point**of**S**if every non-empty neighborhood of*b*intersects**S**and the complement of**S**. The set of all boundary points of**S**is called the boundary of**S**, denoted by**bd**(**S**). - A point
*s*is called**S****interior point**of**S**if there exists a neighborhood of*s*completely contained in**S**. The set of all interior points of**S**is called the interior, denoted by**int**(**S**). - A point
*t*is called**S****isolated point**of**S**if there exists a neighborhood**U**of*t*such that**U****S**= { t }. - A point
*r*is called**S****accumulation point**, if every neighborhood of*r*contains infinitely many distinct points of**S**.