Category:Metric Spaces

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This category contains results about Metric Spaces.
Definitions specific to this category can be found in Definitions/Metric Spaces.


A metric space $M = \struct {A, d}$ is an ordered pair consisting of:

$(1): \quad$ a non-empty set $A$

together with:

$(2): \quad$ a real-valued function $d: A \times A \to \R$ which acts on $A$, satisfying the metric space axioms:
\((M1)\)   $:$     \(\displaystyle \forall x \in A:\) \(\displaystyle d \left({x, x}\right) = 0 \)             
\((M2)\)   $:$     \(\displaystyle \forall x, y, z \in A:\) \(\displaystyle d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right) \)             
\((M3)\)   $:$     \(\displaystyle \forall x, y \in A:\) \(\displaystyle d \left({x, y}\right) = d \left({y, x}\right) \)             
\((M4)\)   $:$     \(\displaystyle \forall x, y \in A:\) \(\displaystyle x \ne y \implies d \left({x, y}\right) > 0 \)             

Subcategories

This category has the following 41 subcategories, out of 41 total.

B

E

F

H

I

L

M

N

O

P

Q

U

Pages in category "Metric Spaces"

The following 133 pages are in this category, out of 133 total.