Topology
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=Topology= | =Topology= | ||
==What is a topology on a set?== | ==What is a topology on a set?== | ||
| - | ; | + | ;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. | ;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. | **Neighborhood of a point. | ||
| - | + | ||
**Exterior point of a set. Exterior of a set. | **Exterior point of a set. Exterior of a set. | ||
**Boundary point of a set. Boundary of a set. | **Boundary point of a set. Boundary of a set. | ||
| Line 19: | Line 21: | ||
**Open sets in the norm-infinity topology. | **Open sets in the norm-infinity topology. | ||
*Fact: The "ball" topology on R<sup>n</sup> is the same as the "box" topology on R<sup>n</sup> | *Fact: The "ball" topology on R<sup>n</sup> is the same as the "box" topology on R<sup>n</sup> | ||
| + | **[[Equivalence of the Box/Ball topology]] | ||
*Induced topology on a subset of R<sup>n</sup> | *Induced topology on a subset of R<sup>n</sup> | ||
Current revision as of 13:18, 10 February 2006
Contents |
[edit] Topology
[edit] What is a topology on a set?
- 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.
- Neighborhood of a point.
- 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.
[edit] The "ball" topology on Rn
- The norm-2 on Rn.
- Open balls in norm-2.
- Open sets in the norm-2 topology.
[edit] 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
