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wisvuze

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thank you

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In summary, the result that every metric equivalent to the standard metric satisfies a well-known theorem does not generalize to arbitrary metric spaces. This can be seen in examples such as the discrete metric, where all sets are open and the open intervals may not be open sets. The open-ness of the intervals is a crucial part for any proof of this theorem.

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wisvuze

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thank you

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Well, every metric equivalent to the standard metric satisfies this. But the result does not generalize to arbitrary metric spaces. For example, take the discrete metric

[tex]d(x,y)=1~\text{if}~x\neq y~\text{and}~d(x,x)=0[/tex]

then all sets are open. In particular, the singletons are open. But the singletons are not the union of open intervals!

It can also happen that the open intervals are not open sets anymore!

- #3

wisvuze

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I should have realized though that the open-ness of the "open interval" depends on the metric ( in particular, one that acts something like the standard metric ); and the open ness of the intervals is the crucial part for any proof of this theorem that I know

Open sets in R are sets that contain all of their interior points. In other words, for every point in an open set, there exists a small enough neighborhood around that point that is also contained within the set.

In R, open sets are represented as the union of open intervals. This means that an open set can be expressed as the combination of multiple open intervals, each of which contains all of its interior points.

The fact that open sets in R can be expressed as the union of open intervals allows for a more flexible and precise way of defining open sets. This representation also allows for a better understanding of open sets and their properties.

Open sets differ from closed sets in that they do not contain any of their boundary points. In other words, there is always a small gap or "hole" around the boundary of an open set, while a closed set includes all of its boundary points.

Yes, open sets can be empty. An open set is considered empty if it does not contain any points at all, including interior points. An example of an empty open set in R would be the set of all real numbers between 1 and 2, including 1 but not 2.

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