Metric Spaces (𝑋,𝑑)
- is a type of mathematical space
- is a tuple (𝑋,𝑑) where:
- 𝑋 be a set of elements
- 𝑑 is a distance metric (i.e. 𝑑 : 𝑋×𝑋 → ℝ+)
- is a type of topological space (𝑋,𝜏) where distance metric (𝑑) is defined and 𝑑 induces a topology (𝜏) on set 𝑋
Metric Space - Examples
- All normed vector spaces are metric spaces
- Euclidean Vector Space (ℝ𝑛) with Euclidean Distance Metric (𝑑). (ℝ𝑛,𝑑) forms a metric space.
- Real Numbers (ℝ) with Absolute Distance Metric (𝑑(𝑥,𝑦) = |𝑦-𝑥|). (ℝ,𝑑) forms a metric space.
- Let 𝐺(𝑉,𝐸,𝑊) be a simple, weighted, undirected graph; with distance metric (𝑑) defined as the shortest path function 𝑑:𝑉×𝑉→ℝ+ (i.e. the length of the shortest path between any two vertices). (𝐺,𝑑) forms a metric space.
Subpages
- Completeness - Non-Completeness
List indent undo
- Metric Spaces (Compactness／Compact - Sequential Compactness - Sequentially Compact - Limit Point Compactness)
- Metric Spaces (Epsilon Balls - Open Sets - Boundary Points／Sets - Closed Sets - Closures - Bounded Sets ／ Boundedness - Unbounded Sets ／ Unboundedness - Totally Bounded Sets ／ Totally Boundedness)
- Metric Spaces (Limit Points - Accumulation Points - Cluster Points)
- Metric Spaces (Neighborhoods)
- Metric Spaces (Sequences - Convergent／Convergence／Converges - Bounded - Cauchy Sequences)
- Transformations Between Metric Spaces