%%visits: 2 A function that maps a coordinate to a Real number, if
the function has the 3 stated properties above Thinking of it as a
“distance function” helps the definition become more natural. ##
intuition ## rigour metric function := let X be a non-empty set, the function
d : X × X → ℝ is
called metric if: d(x, y) > 0 if
x ≠ y and d(x, x) = 0
non-negative d(x, y) = d(y, x)
symmetric d(x, z) ≤ d(x, y) + d(x, z)
Triangle inequality ## exam clinic ## resources tags :math: In “A
problemtext in advanced calculus” it states between x and y should be:
(i) greater than or equal to zero; (ii) greater than zero if x and y are
distinct; (iii) the same as the distance between y and x; and (iv) no
larger than the sum of the distances produced by taking a detour through
a point z. We formalize these conditions to define a “metric” on a set.
## See also [[isometry]] [[symmetry_group]] isometry and symmetry are
similar.
tags: :metric_spaces_and_topologies: