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

Let S be an arbitrary set in the real line R.
  1. A point b R is called 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).
  2. A point s S is called 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).
  3. A point t S is called isolated point of S if there exists a neighborhood U of t such that U S = { t }.
  4. A point r S is called accumulation point, if every neighborhood of r contains infinitely many distinct points of S.
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