Question

what douse it mean for a set to be closed?

Answer 1

In mathematics, the term "closed set" can refer to a concept in different contexts, such as in topology and analysis. Here's a brief explanation in these contexts:

1. **Topology**: A set \( A \) in a topological space is called closed if it contains all its limit points. This means that if a sequence (or net) of points in \( A \) converges to a point \( x \), then \( x \) must also be in \( A \). Another equivalent definition is that the complement of a closed set (the set of all points not in \( A \)) is an open set.

2. **Real Analysis**: In the context of real numbers, a set is closed if it contains all its boundary points. For example, the interval \([a, b]\) (which includes its endpoints) is closed, whereas the interval \((a, b)\) (which does not include its endpoints) is not closed.

3. **Metric Spaces**: In a metric space, a set is closed if its complement is open. Additionally, if a sequence of points within the set converges to a limit, that limit must also be within the set.

In summary, a closed set is characterized by the presence of its limit points or closure properties in the given space.