Metric Spaces and Functions

Questions and Answers

What is a function defined as?

A relation with no restrictions on elements

An operation that combines two or more sets

A relation where each element in set A corresponds to a unique element in set B (correct)

A collection of open subsets of a metric space

What are the properties of a metric space?

d(p,q) > 0 if p ≠ q; d(p,p) = 0, 2) d(p,q) = d(q,p), 3) d(p,q) ≤ d(p,r) + d(r,q)

What is a limit point of a set E?

A point p where every neighborhood of p contains a point q in E such that q ≠ p.

What is an open cover of a set E?

A collection of open subsets of a metric space X that covers set E. Signup and view all the answers

What is an isolated point in a set E?

A point p in E that is not a limit point of E. Signup and view all the answers

What does it mean for a subset K of a metric space X to be compact?

Every open cover of K contains a finite subcover. Signup and view all the answers

How is the distance in R^k defined?

d(x,y) = |x - y| where x, y belong to R^k. Signup and view all the answers

What is the image of a subset E under a function f?

ƒ(E) = {ƒ(x): x ∈ E} Signup and view all the answers

What is the inverse image of a subset G under a function f?

ƒ⁻¹(G) = {x ∈ A: ƒ(x) ∈ G} Signup and view all the answers

What does it mean for a set E to be closed?

Every limit point of E is a point of E. Signup and view all the answers

What is the theorem regarding a compact set K relative to sets Y and X?

K is compact relative to X if and only if K is compact relative to Y. Signup and view all the answers

What is a segment in the context of real numbers?

(a,b) which means all real numbers x such that a < x < b. Signup and view all the answers

What does it mean for a set to be bounded?

The diameter of M, d(M), is finite. Signup and view all the answers

What does a set E with an infinite subset of a compact set K have?

A limit point in K. Signup and view all the answers

What is a neighborhood of a point p in a metric space?

A set N_r(p) consisting of all points q such that d(p,q) < r. Signup and view all the answers

What is the theorem regarding the inverse of a bijection?

If ƒ: A → B is a bijection, then ƒ⁻¹: B → A is also a bijection. Signup and view all the answers

Study Notes

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.

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Description

This quiz covers fundamental concepts of metric spaces, including properties of distances, limit points, and compact sets. It also explores the mapping of functions from one set to another, ensuring comprehension of these mathematical principles.