- #1

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## Main Question or Discussion Point

How does one redefine the metric of a given set, such as the reals? I thought it would be an interesting concept to have a metric defined like so:

[tex]d_X:X^n \times X^n \to \Re[/tex]

[tex](x_1, x_2, x_3, \cdots, x_n), (y_1, y_2, y_3, \cdots, y_n) \mapsto \sum^{n}_{i=1}{\frac{x_i+y_i}{2}}[/tex]

Does it have to be consistent with common sense? For example, does 5 have to be closer to 4 than 0?

Think about how messed up a graphical representation of that would look.

[tex]d_X:X^n \times X^n \to \Re[/tex]

[tex](x_1, x_2, x_3, \cdots, x_n), (y_1, y_2, y_3, \cdots, y_n) \mapsto \sum^{n}_{i=1}{\frac{x_i+y_i}{2}}[/tex]

Does it have to be consistent with common sense? For example, does 5 have to be closer to 4 than 0?

Think about how messed up a graphical representation of that would look.