Current revision as of 13:18, 10 February 2006

[edit] Topology

Interior points of a set: <m>P \\in int(A) </m> if there exists a ball <m>B</m> centered at <m>P</m> with radius r>0 s.t. <m>B \\subset A</m>; [AW: note/question about this on the discussion page]
Open sets: <m>A \\subset R^{n} </m> is open if <m> A = int(A)</m>
Closed sets.: A set <m> A </m> is said to be closed if it's complement <m>R^{n} - A</m> is open.

- 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 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
- Equivalence of the Box/Ball topology
Induced topology on a subset of Rⁿ

@@ Line 1: / Line 1: @@
 =Topology=
 ==What is a topology on a set?==
-;Open sets. :Sets which do not intersect their boundary, that is if <br><m>A \\subset     \\R ^n</m> then A is open if <m> A = int(A)</m> [CZ: but how do you define int(A)?]<br>
+;Interior points of a set: <m>P \\in int(A) </m> if there exists a ball <m>B</m> centered at <m>P</m> with radius r>0 s.t. <m>B \\subset A</m>
-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 \\in A </m> <br> [AW: How about this?]
+:[AW: note/question about this on the discussion page]
+;Open sets: <m>A \\subset R^{n} </m> is open if <m> A = int(A)</m>
 ;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.
@@ Line 21: / Line 21: @@
 **Open sets in the norm-infinity topology.
 *Fact: The &quot;ball&quot; topology on R<sup>n</sup> is the same as the &quot;box&quot; topology on R<sup>n</sup>
+**[[Equivalence of the Box/Ball topology]]
 *Induced topology on a subset of R<sup>n</sup>