MeMoses said:Well I'm lacking on the definition of "closed" and "bounded" which is what I need to get started
That's not what bounded means. In fact, bounded has nothing to do with boundary. A bounded set is one that is fits inside a ball of some finite radius.MeMoses said:Ok so I assume i can prove something is bounded if every neighborhood of point along the edge(ie at r from x0) contains both a point in the ball and outside the ball. I just took that from the boundary point definition, now that doesn't imply something is closed does it?
A closed set is a set that contains all of its limit points. In other words, every sequence of points within the set must have a limit point that is also within the set.
A bounded set is a set that has a finite or defined limit on its size or extent. In other words, there is a finite distance between any two points within the set.
To prove that a set is closed and bounded, you must show that it satisfies both the definitions of closed and bounded. This can be done by showing that all limit points are contained within the set, and that there is a finite distance between any two points within the set.
A set being compact means that it is both closed and bounded. This has important implications in mathematics and physics, as compact sets have nice properties that make them easier to work with and analyze.
Yes, a set can be compact in one space but not in another. This is because the definition of compactness depends on the specific space in which the set is being analyzed. A set may satisfy the definition of compactness in one space, but not in another.