Metric Spaces and Functions

Function

• A function maps each element 'x' from set A to a unique element ƒ(x) in set B.
• Denoted as ƒ: A → B, where A is the domain and B is the codomain.

Properties of a Metric Space

• Distance is positive for distinct points: d(p,q) > 0 if p ≠ q.
• The distance from a point to itself is zero: d(p,p) = 0.
• Distance is symmetric: d(p,q) = d(q,p).
• Triangle inequality holds: d(p,q) ≤ d(p,r) + d(r,q) for any point r in space X.

Limit Points

• A point p is a limit point of set E if every neighborhood around p contains a point q in E, distinct from p.

Open Cover

• An open cover of a set E in a metric space X is a collection of open subsets such that E is contained within the union of those subsets.

Isolated Point

• A point p in set E is isolated if p is not a limit point of E.

Compact Sets

• A subset K of metric space X is compact if every open cover of K has a finite subcover.

Distance in Euclidean Spaces

• In R^k, the distance between two points x and y is defined as d(x,y) = |x - y|.

Image and Range

• The image of a subset E under function ƒ is ƒ(E), which includes all elements that can be reached by mapping elements from E.
• The range of function ƒ is the image of the entire domain: ƒ(A).

Inverse Image

• The inverse image of a subset G under function ƒ is denoted ƒ⁻¹(G), consisting of all elements in A that map into G.

Closed Set

• A set E is closed if it contains all its limit points.

Compactness Relative to Subsets

• If K is a subset of Y, which in turn is a subset of X, K is compact in X if and only if K is compact in Y.

Definition of a Segment

• A segment denoted (a,b) represents all real numbers between a and b, not including a and b.

Open Ball Defined

• The open ball B centered at point x with radius r includes all points y in R^k satisfying |y - x| < r.

Inverse of a Bijection

• If function ƒ: A → B is a bijection, then its inverse ƒ⁻¹: B → A is also a bijection.

Bounded Sets

• A set M is bounded if its diameter d(M) is finite, calculated as d(M) = sup {d(x,y) | x,y ∈ M}.

Infinite Subset in Compact Sets

• A set E containing an infinite subset of a compact set K will have at least one limit point in K.

Neighborhood in Metric Space

• A neighborhood of point p, denoted Nr(p), consists of all points q such that the distance d(p,q) is less than a specified radius r.