I study Rudin's book on analysis. I want to check with you that I understand the definition of closure. Therefore I must check with you that I understand the definition of limit points.
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A neighborhood is for example an interval, a circle or a sphere around a point or a set.
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A limit point is any point that "can be approached as a limit" in contrast to an isolated point for example the set $\{\{1\}, [2,3]\}$ i.e. the union of the point $1$ and the interval between $2$ and $3$ has an isolated point $1$ which is not a limit point but any points between $2$ and $3$ are limit points.
Do you agree?
What I don't understand about closure is: How can there be limit points of a set $E$ which are outside of $E$? I think that the example is an open set but I don't get the picture. I think that the closure by definition always is a closed set and therefore the union of an open set and its limit points is not closed, but I still don't understand how a set can have a limit point not in the set.
I must have misunderstood.
Best Answer
Here's an important point that some of the other answers aren't putting clearly, and I think it's worth stating explicitly.
The phrase "the limit points of the set $A$" doesn't make sense on its own. People will say it, but it's shorthand for something more precise: as it stands, this phrase is missing a crucial piece of information. A correct way of putting this might be "the limit points of the subset $A$ of the topological space $X$", or "the limit points of the set $A$ as a subspace of $X$", or similar.
Limit points are always relative to an ambient space. For example, you can ask the question "what are the limit points of the set $(0,1)$ inside the space $X$?" for each of the following four spaces $X$, and you'll get four different answers: