MetricSpace
Set-Based
Topology
- Definition of Metric Space
- Neighborhood, Limit Points, Openness, Closedness
- Interior, Closure, Boundary
- A Nonempty Perfect Set in Euclidean Space is Uncountable
Compactness
- Compactness
- Several Equivalent Properties of Compactness
- Extreme Value Theorem
- Generalized Cantor’s Intersection Theorem
- Heine-Borel Theorem
- Borel-Lebesgue Theorem
Function-Based
Completeness
Continuity
References
- Walter Rudin, Principles of Mathmatical Analysis (3rd Edition, 1976)
All posts
- Equivalence of Various Compactnesses in Metric Spaces
- In a Metric Space, Compact Implies Closed and Bounded
- Uniform Continuity in Metric Spaces
- Definition of a Metric Space
- Balls and Open Sets, Closed Sets in Metric Spaces
- Completeness and Density in Metric Spaces
- Inner Enclosure Boundary in Metric Spaces
- Topological Isomorphism in Metric Spaces
- Sets Outside/Inside a Certain Distance from the Boundary of a Set
- Polish Space
- Closure and Derived Set in Metric Space
- Neighborhood, Limit Point, Open, Closed in Metric Space
- Properties of Open and Closed Sets in Metric Spaces
- Relatively Open Sets in Metric Spaces
- Closed Subsets of Compact Sets in Metric Spaces are Compact
- Compactness in Metric Space
- Every k Cell is Compact
- Generalized Cantor's Intersection Theorem in Metric Spaces
- Every Non-Empty Perfect Set in Euclidean Space is Uncountable
- Convergence of Cauchy Sequences in Metric Spaces
- Convergence of Sequences in Metric Spaces
- Diameter of a Set in a Metric Space
- Heine-Borel Theorem
- Borel-Cantelli Lemma
- Connected Sets in Metric Spaces
- Continuity and Compactness in Metric Spaces
- Continuous and Uniformly Continuous in Metric Spaces
- Equivalent Conditions for a Function to Be Continuous in a Metric Space
- Limit of Functions in Metric Spaces
- Maximum and Minimum Theorem in Metric Spaces
- Proof that Continuous Functions on Compact Metric Spaces are Uniformly Continuous
- Properties of Complete Metric Spaces
- Properties of Continuous Functions in Metric Spaces
- Properties of Limits of Functions in Metric Spaces
- The Composition of Continuous Functions in Metric Spaces Preserves Continuity
- The Importance of Compactness Conditions in Metric Spaces
- The inverse of a continuous bijection on a compact metric space is continuous.
- Lipschitz Continuity