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Alright, instead of starting a new thread everytime I have a question, I will just post it in here.
Note: These are not from assignments.
Note: Most of these questions can be found in Topology by Munkres. I will make a mention when it is, and where it is.
So, here is the first one...
This is from Munkres, found on page 126.
Define the following metric on R^n as follows...
[tex]d'(x,y) = \sum_{i=1}^{n} \left[ |x_i - y_i|^{p} \right]^{1/p}[/tex]
...where p >= 1.
Show that it induces the usual topology on R^n.
I barely know the first step to showing that if x is in d-ball (usual R^n ball), then there exists a d'-ball that contains x and it is contained in d-ball.
I'll spend the next few minutes thinking about it, or longer. I would certainly ask my prof. tomorrow.
Note: These are not from assignments.
Note: Most of these questions can be found in Topology by Munkres. I will make a mention when it is, and where it is.
So, here is the first one...
This is from Munkres, found on page 126.
Define the following metric on R^n as follows...
[tex]d'(x,y) = \sum_{i=1}^{n} \left[ |x_i - y_i|^{p} \right]^{1/p}[/tex]
...where p >= 1.
Show that it induces the usual topology on R^n.
I barely know the first step to showing that if x is in d-ball (usual R^n ball), then there exists a d'-ball that contains x and it is contained in d-ball.
I'll spend the next few minutes thinking about it, or longer. I would certainly ask my prof. tomorrow.
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