Metric Spaces

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Homework Statement



[PLAIN]http://img833.imageshack.us/img833/6932/metric2.jpg [Broken]

The Attempt at a Solution



I've shown [itex]d_{X\times Y}[/itex] is a metric by using the fact that [itex]d_X[/itex] and [itex]d_Y[/itex] are metrics.

What is a simpler description of [itex]d_N[/itex] with [itex]d_X[/itex] the discrete metric?

Is it just: [tex]d_N(x,y) = \left\{ \begin{array}{lr}
N & : x\neq y\\
0 & : x=y
\end{array}
\right.[/tex]
 
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Answers and Replies

  • #2
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What is a simpler description of [itex]d_N[/itex] with [itex]d_X[/itex] the discrete metric?

Is it just: [tex]d_N(x,y) = \left\{ \begin{array}{lr}
N & : x\neq y\\
0 & : x=y
\end{array}
\right.[/tex]
No. How did you get there??

Just calculate

[tex]d_N((x_1,x_2),(y_1,y_2))[/tex]

and see what the possible outcomes are.
 
  • #3
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No. How did you get there??

Just calculate

[tex]d_N((x_1,x_2),(y_1,y_2))[/tex]

and see what the possible outcomes are.
Yeah I see what I assumed wrong.

[itex]d_N(x,y)[/itex] is the number of coordidates in which x and y differ.

How do I describe these open balls?

The definition is: [itex]B(x,r)=\{ y\in X : d(x,y)<r \}[/itex] where [itex]x\in X[/itex] and [itex]r>0[/itex] is the radius.

So we want:

[itex]B((0,0),1)=\{ y\in X^2 : d((0,0),y)<1 \}[/itex] ;

[itex]B((0,0),2)=\{ y\in X^2 : d((0,0),y)<2 \}[/itex] ;

[itex]B((0,0),3)=\{ y\in X^2 : d((0,0),y)<3 \}[/itex] .
 

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