Definition of a Metric Space

Topology · Axiom Academy

LESSON Definition of a Metric Space Understanding the formal definition of distance and the three axioms that define a metric In mathematics, we often need to talk about distance . How far apart are two points? How close is one number to another? A metric space gives us a rigorous way to answer these questions. A metric space consists of two things: A set (a collection of objects, which we call "points") A function that measures the "distance" between any two points A metric space is an ordered pair where is a set and is a function (called a metric or distance function ) that satisfies three properties for all : These three axioms capture our intuitive understanding of what "distance" should mean. Let's explore each one in detail. The first axiom has two parts: Interpretation: Distances are always non-negative, and the distance from a point to itself is zero. Interpretation: If two points are at zero distance, they must be the same point. Different points must have positive distance between them. Interpretation: The distance from point to point is the same as the distance from to . Interpretation: The direct distance from to is at most the distance of going from to and then from to . This is called the "triangle inequality" because if you visualize three points forming a triangle, it says that any one side is no longer than the sum of the other two sides. Now that we understand the definition, let's look at some concrete examples:

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