What is the Gist of a “Metric Space” or a “Metric” (Mathematics)?

“A metric space is an ordered pair  where  is a set and  is a metric on .”

“The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:

• “the distance from a point to itself is zero,
• “the distance between two distinct points is positive,
• “the distance from A to B is the same as the distance from B to A, and
• “the distance from A to B (directly) is less than or equal to the distance from A to B via any third point C.”

“The function  is also called distance function or simply distance. Often,  is omitted [when describing the metric space symbolically,] and one just writes … if it is clear from the context what metric is used.”

(Quotes selected from “Metric space,” Wikipedia, retrieved 6/4/2020)

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Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).