# 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:

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 75 subcategories, out of 75 total.

### B

- Balls (empty)
- Banach Fixed-Point Theorem (6 P)

### C

- Closed Sets (Metric Spaces) (3 P)
- Completion Theorem (8 P)
- Convergent Mappings (empty)

### D

- Distance Functions (empty)

### E

### F

### H

- Hausdorff Metric (1 P)
- Heine-Cantor Theorem (3 P)

### I

### L

- Lebesgue's Number Lemma (3 P)
- Lipschitz Continuity (2 P)
- Local Boundedness (empty)

### M

- Metric Space is Hausdorff (3 P)
- Metric Space is Paracompact (3 P)
- Metric Space is T4 (3 P)
- Metric Subspaces (9 P)

### N

### O

- Open Sets (Metric Spaces) (1 P)
- Oscillation (5 P)

### P

- Pointwise Equicontinuity (1 P)
- Polish Spaces (empty)

### Q

### R

- Regions (empty)

### S

### T

### U

- Uniform Equicontinuity (1 P)

## Pages in category "Metric Spaces"

The following 153 pages are in this category, out of 153 total.

### C

- Cartesian Metric is Rotation Invariant
- Cauchy Sequence in Metric Space is not necessarily Convergent
- Center is Element of Closed Ball
- Characterization of Convergent Net in Metric Space
- Characterization of Lipschitz Continuity on Shift of Finite Type by Variations
- Characterization of N-Cube
- Closed Ball contains Smaller Closed Ball
- Closed Ball contains Smaller Open Ball
- Closed Real Interval is Closed Set
- Closed Set in Metric Space is G-Delta
- Closure of Non-Empty Bounded Subset of Metric Space is Bounded
- Closure of Subset of Metric Space by Convergent Sequence
- Combination Theorem for Continuous Mappings/Metric Space
- Combination Theorem for Limits of Mappings/Metric Space
- Compact Metric Space is Separable
- Compact Subspace of Metric Space is Sequentially Compact in Itself
- Completeness Criterion (Metric Spaces)
- Completion Theorem (Metric Space)
- Composite of Continuous Mappings at Point between Metric Spaces is Continuous at Point
- Composite of Continuous Mappings between Metric Spaces is Continuous
- Composite of Continuous Mappings on Metric Spaces is Continuous
- Composition of Distance-Preserving Mappings is Distance-Preserving
- Composition of Isometries is Isometry
- Constant Function is Uniformly Continuous/Metric Space
- Continuity of Mapping between Metric Spaces by Closed Sets
- Continuous Extension from Dense Subset
- Continuous Image of Path-Connected Set is Path-Connected/Metric Space
- Continuous Mapping is Continuous on Induced Topological Spaces
- Convergent Sequence in Metric Space has Unique Limit
- Convergent Sequence in Metric Space is Bounded
- Countably Compact Metric Space is Compact
- Countably Compact Metric Space is Sequentially Compact

### D

- User:Dfeuer/Open Set may not be Open Ball
- Diameter of Closure of Subset is Diameter of Subset
- Distance Function for Distinct Elements in Metric Space is Strictly Positive
- Distance Function of Metric Space is Continuous
- Distance-Preserving Image Isometric to Domain for Metric Spaces
- Distance-Preserving Mapping is Injection of Metric Spaces
- Distinct Points in Metric Space have Disjoint Neighborhoods

### E

- Empty Set is Open and Closed in Metric Space
- Empty Set is Open in Metric Space
- Equivalence of Definitions of Convergent Sequence in Metric Space
- Equivalence of Definitions of Topology Induced by Metric
- Euclidean Space is Path-Connected
- Euclidean Space without Origin is Path-Connected
- Existence of Sequence in Subset of Metric Space whose Limit is Infimum
- Extremally Disconnected Metric Space is Discrete
- Extreme Value Theorem

### F

### L

### M

- Metric Defines Norm iff it Preserves Linear Structure
- Metric Induced by a Pseudometric
- Metric Induced by Norm is Invariant Metric
- Metric Induced by Norm is Metric
- Metric Induced by Norm on Normed Division Ring is Metric
- Metric Induces Topology
- Metric is Continous Mapping
- Metric on Shift of Finite Type is Metric
- Metric on Shift of Finite Type is Non-Archimedean
- Metric Space Compact iff Complete and Totally Bounded
- Metric Space Continuity by Epsilon-Delta
- Metric Space defined by Closed Sets
- Metric Space fulfils all Separation Axioms
- Metric Space generates Uniformity
- Metric Space is Closed in Itself
- Metric Space is Compact iff Countably Compact
- Metric Space is Completely Normal
- Metric Space is Countably Compact iff Sequentially Compact
- Metric Space is First-Countable
- Metric Space is Fully Normal
- Metric Space is Fully T4
- Metric Space is Hausdorff
- Metric Space is Lindelöf iff Second-Countable
- Metric Space is Open and Closed in Itself
- Metric Space is Open in Itself
- Metric Space is Paracompact
- Metric Space is Perfectly Normal
- Metric Space is Perfectly T4
- Metric Space is Separable iff Second-Countable
- Metric Space is T4
- Metric Space is T5
- Metric Space is Weakly Countably Compact iff Countably Compact
- Metric Space is Weakly Locally Compact iff Strongly Locally Compact
- Moore-Osgood Theorem

### N

- Neighbourhood of Point Contains Point of Subset iff Distance is Zero
- Non-Archimedean Norm iff Non-Archimedean Metric
- Non-Archimedean Norm iff Non-Archimedean Metric/Necessary Condition
- Non-Archimedean Norm iff Non-Archimedean Metric/Sufficient Condition
- Norm Topology Induced by Metric Induced by Norm
- Null Sequence induces Local Basis in Metric Space
- Null Sequence induces Local Basis in Metric Space/Sequence of Reciprocals
- Null Sequence induces Neighborhood Basis of Closed Sets in Metric Space
- Number Smaller than Lebesgue Number is also Lebesgue Number

### O

### P

- P-adic Norm not Complete on Rational Numbers
- P-Product Metric Induces Product Topology
- Point at Distance Zero from Closed Set is Element
- Pointwise Minimum of Metric and Positive Real Number is Topologically Equivalent Metric
- Positive Multiple of Metric is Metric
- Product of Uniformly Convergent Sequences of Bounded Functions is Uniformly Convergent
- Pseudometric Space is Metric Space iff Kolmogorov

### R

### S

- Second Subsequence Rule
- Separable Metric Space is Second-Countable
- Sequence Converges to Point Relative to Metric iff it Converges Relative to Induced Topology
- Sequence is Bounded in Norm iff Bounded in Metric
- Sequence is Bounded in Norm iff Bounded in Metric/Necessary Condition
- Sequence is Bounded in Norm iff Bounded in Metric/Sufficient Condition
- Sequence of Implications of Metric Space Compactness Properties
- Sequence of Reciprocals induces Local Basis in Metric Space
- Sequential Continuity is Equivalent to Continuity in Metric Space
- Sequential Continuity is Equivalent to Continuity in Metric Space/Corollary
- Sequentially Compact Metric Space is Compact
- Sequentially Compact Metric Space is Lindelöf
- Sequentially Compact Metric Space is Second-Countable
- Sequentially Compact Metric Space is Separable
- Sequentially Compact Metric Space is Totally Bounded
- Set is Closed in Metric Space iff Closed in Induced Topological Space
- Set is Open iff Union of Open Balls
- Sphere is Set Difference of Closed Ball with Open Ball
- Squeeze Theorem/Sequences/Metric Spaces
- Subsequence of Sequence in Metric Space with Limit
- Subset of Metric Space contains Limits of Sequences iff Closed
- Subset of Metric Space is Closed iff contains all Zero Distance Points
- Subspace of Separable Metric Space is Separable

### T

- Three Points in Ultrametric Space have Two Equal Distances
- Three Points in Ultrametric Space have Two Equal Distances/Corollary
- Topology induced by Scaled Euclidean Metric on Positive Integers is Discrete
- Topology induced by Usual Metric on Positive Integers is Discrete
- Totally Bounded Metric Space is Second-Countable
- Totally Bounded Metric Space is Separable

### U

- Uniform Continuity on Metric Space does not imply Compactness
- Uniform Contraction Mapping Theorem
- Uniform Convergence is Hereditary
- Uniform Limit Theorem
- Uniformly Continuous Function is Continuous/Metric Space
- Uniformly Convergent iff Difference Under Supremum Metric Vanishes
- Uniformly Convergent Sequence Evaluated on Convergent Sequence
- Uniformly Convergent Sequence of Bounded Functions is Uniformly Bounded