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metric space
This term is a technical foundation of mathematical analysis and topology. It describes a set where a precise notion of distance is defined, allowing for the formal study of limits, continuity, and convergence. While the most common example is the Euclidean space used in everyday geometry, the concept extends to more abstract settings, such as the distance between two strings of text or the difference between two functions.
In a professional mathematical context, the term is used with high precision to distinguish between general topological spaces (which may only define "closeness") and those that possess a specific numerical distance function. It is almost exclusively used in academic, scientific, or technical registers.
Meanings
Examples
A metric space is a set provided with a distance function.
Does this specific set satisfy the axioms of a metric space?
The real numbers form a complete metric space under the standard absolute value distance.
I wonder if this mapping is an isometry between the two metric spaces.
We need to define a metric space to discuss the convergence of these sequences.
The discrete metric turns any set into a metric space.
Every normed vector space is naturally a metric space.
Let us assume that X is a compact metric space for the purpose of this proof.