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Properties of metric subsets and subspaces - Coggle Diagram
Properties of metric subsets and subspaces
QUALITIES OF SUBSETS:
Boundedness, Path-connectedness, density, open vs. closed subsets, compactness, diameter
BOUNDED SUBSETS:
Criteria:
definition, diameter criterion
Property:
Encapsulation
Examples:
Compact sets
COMPACTNESS
PROPERTIES OF COMPACT SETS:
Bounded
Closed
Completeness
Inheritance theorem for closed subsets
Extreme value theorem
Cantor's theorem
Preservation under continuous maps
EXAMPLES OF COMPACT SETS
General cases:
Singletons, Finite union, Heine-Borel theorem
Specific examples
: Closed intervals in R, Closed unit sphere in Euclidean field space
Counter-examples:
R, Open intervals in R, Unit sphere in C[-1,1], l-infinity with sup norm
OPEN VS. CLOSED SUBSETS:
Criteria:
Definitions, union of open balls, closure-of-limits, complement criterion
Examples of open sets
GENERAL EXAMPLES:
Slice criterion
Trivial sets (empty, metric space),
Open balls
Union of open sets
Finite intersections of open sets
SPECIFIC EXAMPLES:
Open intervals/rays in R
Semi-closed intervals in semi-closed rays
Discrete subsets
Examples of closed sets
SPECIFIC EXAMPLES:
Closed intervals in R
unit circle
example in C[a,b]
Discrete subsets
Subspaces of finite-dim normed spaces
GENERAL EXAMPLES:
Singletons
Trivial sets
Closed balls, spheres,
Finite unions of closed sets
Arbitrary intersections of closed sets
Limit-extension of sequence set
Slice criterion
Completeness criterion
Compact sets
ELEMENTS
Parts of subsets:
interior, closure, boundary of subset:
Criteria:
interior/exterior points, limit criterion, minimality/maximality criterion
Properties:
Openness/Closedness
Set bounds/extremes Duality, Idempotence
Geometrically-intuitive in normed spaces
Examples:
1. Dense subsets
(e.g. Q)
2. Semi-open intervals in R
(closure=closed interval, interior=open interval)
Points in subsets: Interior and exterior points
BALLS
(OPEN, CLOSED, SPHERES)
Properties of open balls:
Symmetry of containment
Coincidence property
Nested ball property
Slice property
Closed ball containment