# 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:

\((M1)\) | $:$ | \(\displaystyle \forall x \in A:\) | \(\displaystyle \map d {x, x} = 0 \) | |||||

\((M2)\) | $:$ | \(\displaystyle \forall x, y, z \in A:\) | \(\displaystyle \map d {x, y} + \map d {y, z} \ge \map d {x, z} \) | |||||

\((M3)\) | $:$ | \(\displaystyle \forall x, y \in A:\) | \(\displaystyle \map d {x, y} = \map d {y, x} \) | |||||

\((M4)\) | $:$ | \(\displaystyle \forall x, y \in A:\) | \(\displaystyle x \ne y \implies \map d {x, y} > 0 \) |

## Subcategories

This category has the following 46 subcategories, out of 46 total.

### B

### C

### D

### E

### F

### H

### I

### L

### M

### N

### O

### P

### Q

### S

### T

### U

## Pages in category "Metric Spaces"

The following 141 pages are in this category, out of 141 total.

### C

- Cartesian Metric is Rotation Invariant
- Center is Element of Closed Ball
- Center is Element of Open Ball
- Characterisation of Cauchy Sequence in Non-Archimedean Norm/Corollary 1
- Characterization of N-Cube
- Closed Ball contains Smaller Closed Ball
- Closed Ball contains Smaller Open Ball
- Closed Ball is Closed
- Closed Real Interval is Closed Set
- Closed Set in Metric Space is G-Delta
- Closure of Subset of Metric Space by Convergent Sequence
- 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 is Continuous/Corollary
- 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
- Continuous Mapping is Continuous on Induced Topological Spaces
- Contraction Theorem
- 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

### E

- Empty Set is Closed in Metric Space
- 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 Metric
- Metric Induced by Norm on Normed Division Ring is Metric
- Metric Induces Topology
- Metric is Continuous
- 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 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
- Null Sequence induces Local Basis in Metric Space
- Null Sequence induces Local Basis in Metric Space/Sequence of Reciprocals
- Number Smaller than Lebesgue Number is also Lebesgue Number

### O

### P

### 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 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

### 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
- Triangle Inequality for Distance from Element to Set

### 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