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Metric Subsets and Subspaces - Coggle Diagram
Metric Subsets and Subspaces
Balls:
spheres, open balls, closed balls
Properties of open balls:
Symmetry of containment
Coincidence property
Nested ball property
Slice property
Closed ball containment
Parts of subsets:
interior, closure, boundary of subset
Criteria for interiors/closures/boundaries:
Interior/exterior point characterization
Limit characterization of closure
Properties of interiors, closures and boundaries of subset:
Openness/Closedness
Set bounds/extremes
Idempotence
Geometrically-intuitive in normed spaces
Examples of interiors and closures:
Dense subsets (closure=metric space)
Q (closure=R, interior=empty)
Semi-open intervals in R (closure=closed interval, interior=open interval)
Qualities of subsets:
Boundedness, Path-connectedness, density, open vs. closed subsets, compactness
Open vs. Closed Sets
Criteria for open/closed sets
Openness:
Union of open balls
Slice criterion
Closedness:
Closure-of-limits criterion
Completeness criterion
Slice criterion
Examples of open and closed sets
Open sets
General cases:
Trivial sets: metric space & empty set
Open balls
Union of open sets
Finite intersection of open sets
Specific examples:
[0,1) in [0, infinity)
Every subset in discrete metric space
Open intervals/rays in R
Closed sets
General cases:
Singletons
Closed balls
Finite unions of closed sets
Arbitrary intersections of closed sets
Limit-extension of sequence set
Specific examples:
Closed intervals in R
Unit circle
Example in C([a,b])
Subspaces of finite-dim normed spaces
Compactness
Criterion for compactness in finite-dim normed space:
Heine-Borel Theorem
Properties of compact spaces/sets:
Bounded
Closed
Completeness
Extreme Value Theorem
Inheritance theorem
Preservation under continuous maps
Cantor's Theorem
Examples of compact/ non-compact sets
General cases of compact sets:
Singletons
Finite union of compact sets
Specific examples of compact sets:
Closed intervals in R
Closed unit sphere in Euclidean field space
Specific examples of non-compact sets:
R
Open intervals in R
Unit sphere in C[-1,1] with max norm
Unit sphere in l-infinity with supremum norm
Bounded subsets
Encapsulation property
Interior and exterior points in subset