Category:Definitions/Metric Spaces

From ProofWiki
Jump to navigation Jump to search

This category contains definitions related to Metric Spaces.
Related results can be found in Category: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:
\((\text M 1)\)   $:$     \(\ds \forall x \in A:\) \(\ds \map d {x, x} = 0 \)      
\((\text M 2)\)   $:$   Triangle Inequality:      \(\ds \forall x, y, z \in A:\) \(\ds \map d {x, y} + \map d {y, z} \ge \map d {x, z} \)      
\((\text M 3)\)   $:$     \(\ds \forall x, y \in A:\) \(\ds \map d {x, y} = \map d {y, x} \)      
\((\text M 4)\)   $:$     \(\ds \forall x, y \in A:\) \(\ds x \ne y \implies \map d {x, y} > 0 \)      

Subcategories

This category has the following 45 subcategories, out of 45 total.

B

C

D

E

F

H

I

L

M

N

O

P

Q

R

S

T

U

Pages in category "Definitions/Metric Spaces"

The following 160 pages are in this category, out of 160 total.

A

B

C

D

E

F

H

I

L

M

N

O

P

Q

R

S

T

U

Retrieved from "https://proofwiki.org/w/index.php?title=Category:Definitions/Metric_Spaces&oldid=700499"