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EDIT: I posted this in the wrong forum, will repost in textbook questions. Please delte this (or move it).
The Q: Define the distance between points \left( x_1 , y_1\right) and \left( x_2 , y_2\right) in the plane to be
\left| y_1 -y_2\right| \mbox{ if }x_1 = x_2 \mbox{ and } 1+ \left| y_1 -y_2\right| \mbox{ if }x_1 \neq x_2 .
Show that this is indeed a metric, and that the resulting metric space is locally compact. I need help with the second part.
My A: Write
d\left( \left( x_1 , y_1\right) , \left( x_2 , y_2\right) \right) = \delta_{x_1}^{x_2} + \left| y_1 -y_2\right| ,
where
\delta_{x_1}^{x_2}=\left\{\begin{array}{cc}0,&\mbox{ if }<br /> x_{1} = x_{2}\\1, & \mbox{ if } x_{1} \neq x_{2}\end{array}\right.
is the Kronecker delta function. Then d:\mathbb{R} ^2 \times \mathbb{R} ^2 \rightarrow \mathbb{R} is a metiric on \mathbb{R} ^2 since the following hold:
i. d is positive definite since d is obviously positive and
\delta_{x_1}^{x_2}=0 \Leftrightarrow x_{1} = x_{2} \mbox{ and } \left| y_1 -y_2\right| = 0 \Leftrightarrow y_{1} = y_{2}
ii. d is symmetric in its variables, that is
d\left( \left( x_1 , y_1\right) , \left( x_2 , y_2\right) \right) = \delta_{x_1}^{x_2} + \left| y_1 -y_2\right| = \delta_{x_2}^{x_1} + \left| y_2 -y_1\right| = d\left( \left( x_2 , y_2\right) , \left( x_1 , y_1\right) \right)
iii. d the triangle inequality, that is: if \left( x_j , y_j\right) \in \mathbb{R} ^2, \mbox{ for } j=1,2,3, then
d\left( \left( x_1 , y_1\right) , \left( x_2 , y_2\right) \right) \leq d\left( \left( x_1 , y_1\right) , \left( x_3 , y_3\right) \right) + d\left( \left( x_3 , y_3\right) , \left( x_2 , y_2\right) \right) ,
which can be reasoned thus: the triangle inequality in R^2 with the Euclidian metric gives
\left| y_1 -y_2\right| \leq \left| y_1 -y_3\right| + \left| y_3 -y_2\right| , \forall y_{1},y_{2},y_{3}\in\mathbb{R}
and
\delta_{x_1}^{x_2} \leq \delta_{x_1}^{x_3} + \delta_{x_3}^{x_2} \mbox{ holds } \forall x_{1},x_{2},x_{3}\in\mathbb{R}
for suppose not: then
\exists x_{1},x_{2},x_{3}\in\mathbb{R} \mbox{ such that }\delta_{x_1}^{x_2} > \delta_{x_1}^{x_3} + \delta_{x_3}^{x_2} \Leftrightarrow x_1 \neq x_2 \mbox{ and } x_1 = x_3 = x_2 ,
which is a contradiction; adding these inequalities yields the required result, viz. the triangle inequality.
By i,ii, and iii, d is a metric on \mathbb{R} ^2.
The locally compact part I don't get: a metric space is locally compact iff every point of has a neighborhood with compact closure.
An open neighborhood of a point, say \left( x_0 , y_0\right), is given by: for some k>0, put
\left\{ \left( x , y\right) : d\left( \left( x , y\right) , \left( x_0 , y_0\right) \right) < k \right\}
but what does that look like? How do I grasp what compact means in this metric?
The delta function above is the discrete metric on R^1 and the absolute value is the Euclidian metric on R^1, and their sum is indeed a metric on the product space R^2. Do I get to keep Heine-Borel? Does Heine-Borel even hold for R^1 with the discrete metric? I don't get the idea of compact sets with H-B, I can tell you "A set is compact if every open cover has a finite subcover," but that topology stuff is so abstract. What does it mean for a set to be compact in terms of a given metric? Is that given by sequential compactness?
Please help with the second part, and let me know if the first is ok.
Thanks,
-Ben
PS: Please don't answer all the questions in the last paragraph, just the ones that help.
The Q: Define the distance between points \left( x_1 , y_1\right) and \left( x_2 , y_2\right) in the plane to be
\left| y_1 -y_2\right| \mbox{ if }x_1 = x_2 \mbox{ and } 1+ \left| y_1 -y_2\right| \mbox{ if }x_1 \neq x_2 .
Show that this is indeed a metric, and that the resulting metric space is locally compact. I need help with the second part.
My A: Write
d\left( \left( x_1 , y_1\right) , \left( x_2 , y_2\right) \right) = \delta_{x_1}^{x_2} + \left| y_1 -y_2\right| ,
where
\delta_{x_1}^{x_2}=\left\{\begin{array}{cc}0,&\mbox{ if }<br /> x_{1} = x_{2}\\1, & \mbox{ if } x_{1} \neq x_{2}\end{array}\right.
is the Kronecker delta function. Then d:\mathbb{R} ^2 \times \mathbb{R} ^2 \rightarrow \mathbb{R} is a metiric on \mathbb{R} ^2 since the following hold:
i. d is positive definite since d is obviously positive and
\delta_{x_1}^{x_2}=0 \Leftrightarrow x_{1} = x_{2} \mbox{ and } \left| y_1 -y_2\right| = 0 \Leftrightarrow y_{1} = y_{2}
ii. d is symmetric in its variables, that is
d\left( \left( x_1 , y_1\right) , \left( x_2 , y_2\right) \right) = \delta_{x_1}^{x_2} + \left| y_1 -y_2\right| = \delta_{x_2}^{x_1} + \left| y_2 -y_1\right| = d\left( \left( x_2 , y_2\right) , \left( x_1 , y_1\right) \right)
iii. d the triangle inequality, that is: if \left( x_j , y_j\right) \in \mathbb{R} ^2, \mbox{ for } j=1,2,3, then
d\left( \left( x_1 , y_1\right) , \left( x_2 , y_2\right) \right) \leq d\left( \left( x_1 , y_1\right) , \left( x_3 , y_3\right) \right) + d\left( \left( x_3 , y_3\right) , \left( x_2 , y_2\right) \right) ,
which can be reasoned thus: the triangle inequality in R^2 with the Euclidian metric gives
\left| y_1 -y_2\right| \leq \left| y_1 -y_3\right| + \left| y_3 -y_2\right| , \forall y_{1},y_{2},y_{3}\in\mathbb{R}
and
\delta_{x_1}^{x_2} \leq \delta_{x_1}^{x_3} + \delta_{x_3}^{x_2} \mbox{ holds } \forall x_{1},x_{2},x_{3}\in\mathbb{R}
for suppose not: then
\exists x_{1},x_{2},x_{3}\in\mathbb{R} \mbox{ such that }\delta_{x_1}^{x_2} > \delta_{x_1}^{x_3} + \delta_{x_3}^{x_2} \Leftrightarrow x_1 \neq x_2 \mbox{ and } x_1 = x_3 = x_2 ,
which is a contradiction; adding these inequalities yields the required result, viz. the triangle inequality.
By i,ii, and iii, d is a metric on \mathbb{R} ^2.
The locally compact part I don't get: a metric space is locally compact iff every point of has a neighborhood with compact closure.
An open neighborhood of a point, say \left( x_0 , y_0\right), is given by: for some k>0, put
\left\{ \left( x , y\right) : d\left( \left( x , y\right) , \left( x_0 , y_0\right) \right) < k \right\}
but what does that look like? How do I grasp what compact means in this metric?
The delta function above is the discrete metric on R^1 and the absolute value is the Euclidian metric on R^1, and their sum is indeed a metric on the product space R^2. Do I get to keep Heine-Borel? Does Heine-Borel even hold for R^1 with the discrete metric? I don't get the idea of compact sets with H-B, I can tell you "A set is compact if every open cover has a finite subcover," but that topology stuff is so abstract. What does it mean for a set to be compact in terms of a given metric? Is that given by sequential compactness?
Please help with the second part, and let me know if the first is ok.
Thanks,
-Ben
PS: Please don't answer all the questions in the last paragraph, just the ones that help.
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