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Hello, I know one proof of this well known theorem that assumes on the metric of R being the standard metric. Does this result generalize to arbitrary metrics on R?
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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.