Show R^2 is locally compact with non-standard metric: I

In summary, the conversation discusses the definition of distance between points in a plane, and shows that it is a metric and that the resulting metric space is locally compact. The conversation also brings up questions about what it means for a set to be compact in terms of a given metric. The answer suggests drawing a picture of the open ball about a point and using concepts such as Heine-Borel and total boundedness to show that every point has a complete and totally bounded neighborhood.
  • #1
benorin
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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 [itex]\left( x_1 , y_1\right) [/itex] and [itex]\left( x_2 , y_2\right) [/itex] in the plane to be

[tex]\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 .[/tex]

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

[tex]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| ,[/tex]

where

[tex]\delta_{x_1}^{x_2}=\left\{\begin{array}{cc}0,&\mbox{ if }
x_{1} = x_{2}\\1, & \mbox{ if } x_{1} \neq x_{2}\end{array}\right.[/tex]

is the Kronecker delta function. Then [itex]d:\mathbb{R} ^2 \times \mathbb{R} ^2 \rightarrow \mathbb{R}[/itex] is a metiric on [itex]\mathbb{R} ^2[/itex] since the following hold:

i. d is positive definite since d is obviously positive and

[tex]\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}[/tex]

ii. d is symmetric in its variables, that is

[tex]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)[/tex]

iii. d the triangle inequality, that is: if [itex]\left( x_j , y_j\right) \in \mathbb{R} ^2, \mbox{ for } j=1,2,3,[/itex] then

[tex]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) ,[/tex]

which can be reasoned thus: the triangle inequality in R^2 with the Euclidian metric gives

[tex]\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}[/tex]

and

[tex]\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}[/tex]

for suppose not: then

[tex]\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 ,[/tex]

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 [itex]\mathbb{R} ^2[/itex].

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 [itex]\left( x_0 , y_0\right) [/itex], is given by: for some k>0, put

[tex]\left\{ \left( x , y\right) : d\left( \left( x , y\right) , \left( x_0 , y_0\right) \right) < k \right\}[/tex]

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.
 
Last edited:
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  • #2
Draw yourself a picture of the open ball about (x0, y0) of radius k. I know you asked what it looks like, but you should be able to easily draw this for yourself. Note that there's a drastic change in what these sets look like when k < ? vs. when k > ?. What is the "?"? Well you should be able to figure that out easily as well. Once you do it, there should be an obvious choice (or range of choices) for k. Then I think you might use something like Heine-Borel.
 
  • #3
in any metric space, compact is equivalent to complete and totally bounded. so just show every point ahs a complete totally bounded nbhd.

i am a little schmooked right now but i think totally bounded means every bounded set has a finite open cover by sets of diameter less than e, for any e>0.
 

1. What is the definition of "locally compact" in mathematics?

Locally compact is a property of a topological space, which means that every point in the space has a neighborhood that is compact. In other words, the space is "locally" compact at every point.

2. How is the R^2 space defined with a non-standard metric?

The R^2 space, also known as the Euclidean plane, is usually defined with the standard metric, which measures the distance between two points using the Pythagorean theorem. However, a non-standard metric can be defined by using a different formula to measure the distance between points. This can result in a different topology for the space.

3. What does it mean for a space to be compact?

A compact space is a topological space in which every open cover has a finite subcover. In simpler terms, this means that the space is "small enough" to be covered by a finite number of open sets. This is a desirable property in mathematics, as compact spaces have many useful properties and simplifies proofs.

4. How is the R^2 space with the non-standard metric shown to be locally compact?

To show that the R^2 space with a non-standard metric is locally compact, we need to prove that every point in the space has a compact neighborhood. This can be done by constructing a compact set around each point that satisfies the definition of a neighborhood. A rigorous proof would involve using the properties of compact sets and the non-standard metric to show that this neighborhood is indeed compact.

5. What are the practical applications of studying locally compact spaces?

Locally compact spaces have many applications in mathematics, physics, and engineering. For example, they are used in differential geometry to study smooth manifolds, in functional analysis to study topological vector spaces, and in quantum mechanics to describe the behavior of particles in a finite space. Understanding the properties of locally compact spaces also helps in solving problems in optimization, control theory, and data analysis.

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