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Understanding limit points is fundamental in topology and analysis. A limit point of a set is a point such that any open interval around it contains other points from the set. In simpler terms, if you can draw a circle (no matter how small) around a limit point and always include some points from the set within this circle, then you've identified a limit point. This concept is particularly important when determining the 'completeness' of a set.

For instance, in the given exercise, we consider the set \(A = \{x: |x| < 1\}\). While the numbers -1 and 1 are not included in set A, they are still limit points because any interval you take around these points will contain members from the set A. To illustrate, let's imagine a tiny interval around -1, such as \((-1.01, -0.99)\). This interval still contains numbers from set A, though -1 itself is not in A. Hence, -1 (and similarly 1) is a limit point of the set, which makes it essential in finding the closure \(\bar{A}\).

When it comes to understanding intervals, it's important to distinguish between open and closed types. A closed interval includes its endpoints, meaning if you have a set that is described by a closed interval, every number between and including those endpoints belongs to the set.

Take the set \(B = \{x: |x| \leq 1\}\) from our example. Represented as a closed interval \([-1, 1]\), it includes all the numbers from -1 to 1. The boundary points -1 and 1 are part of the set, which is indicated by the use of the 'less than or equal to' \(\leq\) in the definition. The significance of closed intervals lies in their 'completeness' – they contain all limit points, which is why they are called 'closed'. This is precisely why, when we talk about the closure of a set, we end up with a closed interval: it is the smallest closed set that includes all the points from the original set along with its limit points.

Now, let's shed light on open intervals. An open interval, such as the one in set \(A = \{x: |x| < 1\}\), includes all the numbers between two endpoints but does not include the endpoints themselves. It can be visualized as a line segment with hollow circles at the ends and is usually denoted by parentheses, like \((-1, 1)\).

Open intervals are 'open' because they don't contain their boundary points. In our example, -1 and 1 are not part of set A. This makes open intervals fundamentally different from closed intervals. When you're working with open intervals in problems, keep in mind that even though the endpoint numbers are not included in the set, they could still play a crucial role as limit points, which we saw when discussing the closure of set A.