From Vectorcalcumb

Revision as of 23:01, 8 February 2006

Topology

What is a topology on a set?

Open sets. 

Sets which do not intersect their boundary, that is if
<m>A \\subset \\R ^n</m> then A is open if <m> A = int(A)</m> [CZ: but how do you define int(A)?]

A is open if for every <m>P \\in A </m> there exists a ball <m> B </m>, with radius > 0, centered at <m> P </m> s.t. <m> B \\subset A </m>
[AW: How about this?]

Closed sets. 

A set <m> A </m> is said to be closed if it's complement <m>R^{n} - A</m> is open.

• • 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