Topology

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<p align="center">Topology</p>
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=Topology=
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<ul>
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==What is a topology on a set?==
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<li>What is a topology on a set?
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;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>
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<ul>
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:[AW: note/question about this on the discussion page]
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<li>Open sets.</li>
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;Open sets: <m>A \\subset R^{n} </m> is open if <m> A = int(A)</m>
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<p>Sets which do not contain their boundary</p>
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;Closed sets. : A set <m> A </m> is said to be closed if it's complement <m>R^{n} - A</m> is open.
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<li>Closed sets.</li>
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**Neighborhood of a point.
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<li>Neighborhood of a point.</li>
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<li>Interior points of a set. Interior of a set.</li>
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**Exterior point of a set. Exterior of a set.
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<li>Exterior point of a set. Exterior of a set.</li>
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**Boundary point of a set. Boundary of a set.
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<li>Boundary point of a set. Boundary of a set.</li>
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**Compact sets.
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<li>Compact sets.</li>
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**Acumulation point of a set.
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<li>Acumulation point of a set.</li>
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</ul><li>
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==The &quot;ball&quot; topology on R<sup>n</sup>==
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<li>The &quot;ball&quot; topology on R<sup>n</sup></li>
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**The norm-2 on R<sup>n</sup>.
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<ul>
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**Open balls in norm-2.
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<li>The norm-2 on R<sup>n</sup>.</li>
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**Open sets in the norm-2 topology.
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<li>Open balls in norm-2.</li>
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==The &quot;box&quot; topology on R<sup>n</sup>==
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<li>Open sets in the norm-2 topology.</li>
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**The norm-infinity on R<sup>n</sup>.
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</ul>
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**Open balls in norm-infinity.
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<li>The &quot;box&quot; topology on R<sup>n</sup></li>
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**Open sets in the norm-infinity topology.
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<ul>
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*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>
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<li>The norm-infinity on R<sup>n</sup>.</li>
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**[[Equivalence of the Box/Ball topology]]
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<li>Open balls in norm-infinity.</li>
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*Induced topology on a subset of R<sup>n</sup>
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<li>Open sets in the norm-infinity topology.</li>
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</ul>
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<li>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></li>
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<li>Induced topology on a subset of R<sup>n</sup></li>
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</ul>
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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
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