Topology

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(Difference between revisions)
(What is a topology on a set?)
(What is a topology on a set?)
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=Topology=
=Topology=
==What is a topology on a set?==
==What is a topology on a set?==
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;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>
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;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>
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;Closed sets. : A set <m> A </m> is said to be closed if it's complement <m>\\overline{A}</m> is open.
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;Closed sets. : A set <m> A </m> is said to be closed if it's complement <m>R<sup>n</sup> \\{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.

Revision as of 11:45, 8 February 2006

Contents

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>
Closed sets. 
A set <m> A </m> is said to be closed if it's complement <m>Rn \\{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
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