Chapter 1 Schaums Flashcards
(17 cards)
Concept
Definition
Neighborhoods
A neighborhood of a point z₀ is the set of all points z such that |z - z₀| < δ. A deleted neighborhood excludes z₀: 0 < |z - z₀| < δ.
Limit Points
A point z₀ is a limit point of a set S if every deleted neighborhood of z₀ contains points of S.
Closed Sets
A set S is closed if it contains all its limit points.
Bounded Sets
A set S is bounded if there exists a constant M such that |z| < M for every point in S. A set that is both bounded and closed is called compact.
Interior, Exterior, and Boundary Points
Interior: neighborhood inside S. Exterior: neighborhood outside S. Boundary: neighborhoods include both.
Open Sets
A set is open if all its points are interior points.
Connected Sets
A set is connected if any two points can be joined by a continuous path inside the set.
Open Regions or Domains
An open connected set is called an open region or domain.
Closure of a Set
The closure of a set includes all its points and its limit points. It is always closed.
Closed Regions
An open region that includes its boundary is a closed region.
Union and Intersection
Union: all points in either set. Intersection: all points in both sets.
Complement of a Set
All points not in a set S. Denoted Sᶜ.
Null Sets and Subsets
Null set ∅ has no elements. A subset contains only points from another set.
Countability of a Set
A set is countable if its elements match the natural numbers. Otherwise, it’s uncountable.
Weierstrass–Bolzano Theorem
Every bounded infinite set has at least one limit point.
Heine–Borel Theorem
If a bounded closed set is covered by open sets, a finite number of those open sets can still cover it.
