HallsofIvy said:I don't know what you mean by "[itex]k_0A[/itex]". Multiplication isn't defined in a general metric space.
A nowhere dense set is a subset of a metric space that has no interior points. This means that no matter how small of a neighborhood you choose within the set, there will always be points outside of that neighborhood. In other words, the set is "thin" and does not contain any isolated points.
A dense set is a subset of a metric space that has points arbitrarily close to every point in the space. This means that every neighborhood in the space contains at least one point from the dense set. On the other hand, a nowhere dense set does not have any points in its interior and therefore cannot be dense.
No, a nowhere dense set cannot contain isolated points. This is because an isolated point would have a neighborhood that does not contain any other points, contradicting the definition of a nowhere dense set.
Nowhere dense sets are often used in the study of topology and functional analysis. They provide a way to classify different types of subsets in a metric space and can help prove theorems about other types of sets, such as dense sets and meager sets.
No, there is a third type of subset known as a meager set. A meager set is a subset of a metric space that can be expressed as the countable union of nowhere dense sets. In other words, it is a "small" set that can be constructed from nowhere dense sets.