"Qualitative Intrinsic Observations.." what?

[nezenic]

I am not sure if this is the correct area, and I am sorry if this isn't the typical type of answer someone would seek when posting on this forum, but I am just confused and need some ideas... My professor is asking a question that I am just having a hard time answering correctly.

The topic is metric spaces. It is pretty straightforward to define the xy-plane as a space with the metric being the usual euclidean distance function. But then I came up with some examples as taking subsets of this set. One of them was to use functions as restrictions as to where x and y can be defined and such (lines, parabolas, circles, etc..). One in particular that was given to me is the "union of the portion of each axis from -1 to 1" which is a "cross" centered at the origin.

Now the question I was given was: What are the qualitative intrinsic differences between this example and the entire plane? And their differences from each other?

I have tried to write quite a bit in answering this question, such as explaining some of the bounds that are enforced, different shapes that can created, continuity, but I seem to just not be answering the problem right..

Any ideas on what kinds of things I am doing wrong here?

 

[pka]

I know a lot about metric spaces.
But I have absolutely no idea about what you wrote.
Please, just state the question exactly as it was given to you to answer.

 

[nezenic]

Ok sorry..

First: Using the Euclidean plane as a metric space, let \(\displaystyle R^2\) be a plane with the coordinates \(\displaystyle (x, y)\) for each point. And the distance between two points \(\displaystyle a_1 = (x_1, y_1)\) and \(\displaystyle a_2 = (x_2, y_2)\) is defined by \(\displaystyle d(a_1, a_2) = ((x_1 - x_2)^2 + (y_1 - y_2)^2)^{1/2}\).

His exact phrasing of his question is: "My example is the set $Y$ built as the union of the portion of each coordinate axis between -1 and 1, inclusive. So the set looks like a cross in the plane. Give this set the “induced metric.” How is this metric space qualitatively different from the Cartesian plane?"

And after I came up with a couple more examples of subsets and tried to answer the question, he replied: "Can you tell me some qualitative “intrinsic” ways that these metric spaces you made are different from the plane and from each other?"

I tried to mention when the metric function is greatest and what happens when certain restrictions are placed on the ordered pairs. But I would not be sure what to say here other than that the cross example is "smaller".

 

[pka]

So the new space is \(\displaystyle \mathcal{S} = \left\{ {(x,y):\left[ {x = 0 \vee y = 0} \right] \wedge \left| x \right| \leqslant 1 \wedge \left| y \right| \leqslant 1} \right\}\).
The space \(\displaystyle \mathcal{S}\) is bounded. Why?
Is the space \(\displaystyle \mathcal{S}\) compact?
Does the space \(\displaystyle \mathcal{S}\) have infinitely many cut-points?
e.t.c.

 

[nezenic]

Ok that helps a lot. Thank you for the help!