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-	<p align="center">Topology</p>	+	=Topology=
-	<ul>	+	==What is a topology on a set?==
-	<li>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>
-	<ul>	+	:[AW: note/question about this on the discussion page]
-	<li>Open sets.</li>	+	;Open sets: <m>A \\subset R^{n} </m> is open if <m> A = int(A)</m>
-	<li>Closed sets.</li>	+	;Closed sets. : A set <m> A </m> is said to be closed if it's complement <m>R^{n} - A</m> is open.
-	<li>Neighborhood of a point.</li>	+	**Neighborhood of a point.
-	<li>Interior points of a set. Interior of a set.</li>	+
-	<li>Exterior point of a set. Exterior of a set.</li>	+	**Exterior point of a set. Exterior of a set.
-	<li>Boundary point of a set. Boundary of a set.</li>	+	**Boundary point of a set. Boundary of a set.
-	<li>Compact sets.	+	**Compact sets.
-	</ul><li>	+	**Acumulation point of a set.
-	<li>The &quot;ball&quot; topology on R<sup>n</sup></li>	+
-	<ul>	+	==The &quot;ball&quot; topology on R<sup>n</sup>==
-	<li>The norm-2 on R<sup>n</sup>.</li>	+	**The norm-2 on R<sup>n</sup>.
-	<li>Open balls in norm-2.</li>	+	**Open balls in norm-2.
-	<li>Open sets in the norm-2 topology.</li>	+	**Open sets in the norm-2 topology.
-	</ul>	+	==The &quot;box&quot; topology on R<sup>n</sup>==
-	<li>The &quot;box&quot; topology on R<sup>n</sup></li>	+	**The norm-infinity on R<sup>n</sup>.
-	<ul>	+	**Open balls in norm-infinity.
-	<li>The norm-infinity on R<sup>n</sup>.</li>	+	**Open sets in the norm-infinity topology.
-	<li>Open balls in norm-infinity.</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>Open sets in the norm-infinity topology.</li>	+	**[[Equivalence of the Box/Ball topology]]
-	</ul>	+	*Induced topology on a subset of R<sup>n</sup>
-	</ul>	+

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Current revision as of 13:18, 10 February 2006

Contents 1 Topology 1.1 What is a topology on a set? 1.2 The "ball" topology on Rn 1.3 The "box" topology on Rn

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