Category:Definitions/Metric Spaces
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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:
together with:
- $(2): \quad$ a real-valued function $d: A \times A \to \R$ which acts on $A$, satisfying the metric space axioms:
| \((\text M 1)\) | $:$ | \(\displaystyle \forall x \in A:\) | \(\displaystyle \map d {x, x} = 0 \) | |||||
| \((\text M 2)\) | $:$ | \(\displaystyle \forall x, y, z \in A:\) | \(\displaystyle \map d {x, y} + \map d {y, z} \ge \map d {x, z} \) | |||||
| \((\text M 3)\) | $:$ | \(\displaystyle \forall x, y \in A:\) | \(\displaystyle \map d {x, y} = \map d {y, x} \) | |||||
| \((\text M 4)\) | $:$ | \(\displaystyle \forall x, y \in A:\) | \(\displaystyle x \ne y \implies \map d {x, y} > 0 \) |
Subcategories
This category has the following 24 subcategories, out of 24 total.
B
C
D
E
F
I
L
N
O
P
Q
T
Pages in category "Definitions/Metric Spaces"
The following 153 pages are in this category, out of 153 total.
B
C
- Definition:Cauchy Equivalent Metrics
- Definition:Cauchy Sequence (Metric Space)
- Definition:Cauchy Sequence in Metric Space
- Definition:Cauchy Sequence/Metric Space
- Definition:Center of Open Ball
- Definition:Closed Ball
- Definition:Closed Ball/Radius
- Definition:Closed Set (Metric Space)
- Definition:Closed Set of Metric Space
- Definition:Closed Set/Metric Space
- Definition:Closed Set/Metric Space/Definition 1
- Definition:Closed Set/Metric Space/Definition 2
- Definition:Closure (Metric Space)
- Definition:Compact Metric Space
- Definition:Complete Metric Space
- Definition:Completely Metrizable Topology
- Definition:Congruence (Metric Spaces)
- Definition:Continuity/Metric Subspace
- Definition:Continuous at Point of Metric Space
- Definition:Continuous Mapping (Metric Space)
- Definition:Continuous Mapping (Metric Space)/Point
- Definition:Continuous Mapping (Metric Space)/Point/Definition 1
- Definition:Continuous Mapping (Metric Space)/Point/Definition 2
- Definition:Continuous Mapping (Metric Space)/Point/Definition 3
- Definition:Continuous Mapping (Metric Space)/Point/Definition 4
- Definition:Continuous Mapping (Metric Space)/Space
- Definition:Continuous Mapping (Metric Space)/Space/Definition 1
- Definition:Continuous Mapping (Metric Space)/Space/Definition 2
- Definition:Continuous on Metric Space
- Definition:Contraction Mapping
- Definition:Convergent Mapping/Metric Space
- Definition:Convergent Sequence/Metric Space
- Definition:Convergent Series/Normed Vector Space
- Definition:Convergent Series/Normed Vector Space/Definition 1
D
- Definition:Diameter of Bounded Metric Subspace
- Definition:Diameter of Subset of Metric Space
- Definition:Disk
- Definition:Distance Between Points
- Definition:Distance from Subset
- Definition:Distance Function
- Distance-Preserving Image Isometric to Domain for Metric Spaces
- Definition:Distance-Preserving Mapping
- Definition:Distance/Points
- Definition:Distance/Sets/Metric Spaces
H
- Definition:Homeomorphic Metric Spaces
- Definition:Homeomorphism
- Definition:Homeomorphism (Metric Spaces)
- Definition:Homeomorphism/Metric Spaces
- Definition:Homeomorphism/Metric Spaces/Definition 1
- Definition:Homeomorphism/Metric Spaces/Definition 2
- Definition:Homeomorphism/Metric Spaces/Definition 3
- Definition:Homeomorphism/Metric Spaces/Definition 4
- Definition:Hölder Continuous
I
L
- Definition:Lebesgue Number
- Definition:Limit of Function
- Definition:Limit of Function (Metric Space)
- Definition:Limit of Function (Metric Space)/Epsilon-Ball Condition
- Definition:Limit of Function (Metric Space)/Epsilon-Delta Condition
- Definition:Limit of Sequence/Metric Space
- Definition:Limit Point/Metric Space
- Definition:Lipschitz Condition
- Definition:Lipschitz Constant
- Definition:Lipschitz Continuity
- Definition:Lipschitz Continuity/Lipschitz Constant
- Definition:Lipschitz Continuous
- Definition:Lipschitz Equivalence
- Definition:Locally Bounded
- Definition:Locally Bounded/Family of Mappings
- Definition:Locally Bounded/Mapping
M
- Definition:Metric
- Definition:Metric Equivalence
- Definition:Metric Induced by Norm
- Definition:Metric Induced by Norm on Division Ring
- Definition:Metric Space
- Definition:Metric Space Axioms
- Definition:Metric Space/Distance Function
- Definition:Metric Space/Point
- Definition:Metric Space/Triangle Inequality
- Definition:Metric Subspace
- Definition:Metrizable Space
- Definition:Metrizable Topology
- Definition:Metrizable Uniformity
N
O
P
- Definition:Path-Connected Metric Space
- Definition:Path-Connected Metric Subspace
- Definition:Path-Connected/Metric Space
- Definition:Path-Connected/Metric Space/Subset
- Definition:Pointwise Equicontinuous
- Definition:Polish Space
- Definition:Projection (Mapping Theory)
- Definition:Pseudometrizable Uniformity
R
S
T
- Definition:Topological Equivalence
- Definition:Topologically Equivalent Metric Spaces
- Definition:Topologically Equivalent Metrics
- Definition:Topologically Equivalent Metrics/Definition 1
- Definition:Topologically Equivalent Metrics/Definition 2
- Definition:Topology Induced by Metric
- Definition:Topology Induced by Metric/Definition 1
- Definition:Topology Induced by Metric/Definition 2
- Definition:Totally Bounded Metric Space
- Definition:Triangle Inequality
U
- Definition:Ultrametric Space
- Definition:Unbounded Metric Space
- Definition:Underlying Set of Metric Space
- Definition:Underlying Set/Metric Space
- Definition:Uniform Continuity
- Definition:Uniform Continuity/Metric Space
- Definition:Uniform Contraction Mapping
- Definition:Uniform Convergence/Metric Space
- Definition:Uniform Equivalence
- Definition:Uniform Equivalence/Metric Spaces
- Definition:Uniform Equivalence/Metrics
- Definition:Uniformly Bounded
- Definition:Uniformly Equicontinuous
