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If (X, d) and (X, r) are metric space, is {X, max(d, r)} necessary a metric space? what about (X, min(d, r))?
ag2ie said:Thanks ...and I think (X, min(d, r)) is not a metric space..right?
No, a Metric Space can be defined as a set of any objects as long as there is a distance function that satisfies the properties of a metric.
The distance function in a Metric Space must satisfy the properties of non-negativity, symmetry, and the triangle inequality.
Yes, a Metric Space can have multiple distance functions as long as they all satisfy the properties of a metric.
The distance between two points in a Metric Space is calculated using the distance function, which takes the two points as input and returns a value that represents the distance between them.
Yes, a Metric Space can have an infinite number of points as long as the distance function is well-defined for all pairs of points.