Topology

From Vectorcalcumb

Topology

• What is a topology on a set?
  • Open sets.

  Sets which do not contain their boundary

  • Closed sets.
  • Neighborhood of a point.
  • Interior points of a set. Interior of a set.
  • Exterior point of a set. Exterior of a set.
  • Boundary point of a set. Boundary of a set.
  • Compact sets.
  • Acumulation point of a set.
• The "ball" topology on Rⁿ
• The norm-2 on Rⁿ.
• Open balls in norm-2.
• Open sets in the norm-2 topology.
• The "box" topology on Rⁿ
• The norm-infinity on Rⁿ.
• Open balls in norm-infinity.
• Open sets in the norm-infinity topology.
• Fact: The "ball" topology on Rⁿ is the same as the "box" topology on Rⁿ
• Induced topology on a subset of Rⁿ