section_path: Important concepts/Metric spaces
section_name: Metric spaces
paragraph_text: Metric spaces
  
  In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.
  
  Much of analysis happens in some metric space; the most commonly used are the real line, the complex plane, Euclidean space, other vector spaces, and the integers. Examples of analysis without a metric include measure theory (which describes size rather than distance) and functional analysis (which studies topological vector spaces that need not have any sense of distance).
  
  Formally, a metric space is an ordered pair (M, d) {\displaystyle (M,d)} where M {\displaystyle M} is a set and d {\displaystyle d} is a metric on M {\displaystyle M}, i.e., a function
  
  such that for any x, y, z ∈ M {\displaystyle x,y,z\in M}, the following holds:
  
  By taking the third property and letting z = x {\displaystyle z=x}, it can be shown that d (x, y) ≥ 0 {\displaystyle d(x,y)\geq 0} (non-negative).