Topology
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=Topology= | =Topology= | ||
==What is a topology on a set?== | ==What is a topology on a set?== | ||
| - | ;Open sets. :Sets which do not contain their boundary, that is if | + | ;Open sets. :Sets which do not contain their boundary, that is if <br><m>A \\subset \\R ^n</m> then A is open if <m> A = int(a)</m> |
| - | <m>A \\subset \\ | + | ;Closed sets. : A set <m> A </m> is said to be closed if it's complement <m>\\overline{A}</m> is open. |
| - | + | ||
**Neighborhood of a point. | **Neighborhood of a point. | ||
**Interior points of a set. Interior of a set. | **Interior points of a set. Interior of a set. | ||
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**Compact sets. | **Compact sets. | ||
**Acumulation point of a set. | **Acumulation point of a set. | ||
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==The "ball" topology on R<sup>n</sup>== | ==The "ball" topology on R<sup>n</sup>== | ||
**The norm-2 on R<sup>n</sup>. | **The norm-2 on R<sup>n</sup>. | ||
Revision as of 05:16, 8 February 2006
Contents |
Topology
What is a topology on a set?
- Open sets.
- Sets which do not contain their boundary, that is if
<m>A \\subset \\R ^n</m> then A 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>\\overline{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
