how to denote the distance of a point x to a set S?

The distance of a point \( x \) to a set \( S \) in a metric space is typically denoted by \( \text{dist}(x, S) \). This distance is defined as the infimum (or the greatest lower bound) of the distances from the point \( x \) to all points in the set \( S \). Mathematically, it can be expressed as:

\[
\text{dist}(x, S) = \inf_{y \in S} d(x, y)
\]

where \( d(x, y) \) is the distance between points \( x \) and \( y \) in the metric space.

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### Expanding Questions:
1. How is the concept of distance between a point and a set used in optimization problems?
2. Can you provide an example where the distance from a point to a set is calculated in a specific metric space?
3. What are the properties of the distance function \( \text{dist}(x, S) \)?
4. How does the distance from a point to a set change if the set is open versus closed?
5. What is the Hausdorff distance, and how is it related to the distance of a point to a set?

### Tip:
When dealing with non-convex sets, finding the exact point in \( S \) that minimizes the distance to \( x \) can be challenging, often requiring numerical methods.

Learn how to denote the distance of a point x to a set S in metric spaces with detailed explanations and mathematical formulas. Explore the concept of infimum and distance functions in the context of metric spaces.

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