Mind-Warm-Ups Vector Spaces

Ultimate Space = S
si = arbitrary partition of S
Δsi = infinitesimal si, that is limit of si as it tends to zero.
pi = arbitrary point in S = Δsi

Problem 31: How are the following spaces different from each other:
(i) A linear space and a vector space
(ii) A vector space and an Euclidean space
(iii) An Euclidean space and a metric space
Ans: (i) a linear space is also called a vector space
(ii) An Euclidean space is a vector space in which the distance between points p1 = (α1,...,αn) and p2 = (β1,...,βn is defined as
|p1 - p2| = [(α11)2 + (α22)2 + ... + (αnn)2]1/2, where α and β are in the real field R. This Euclidean space is an n-dimensional Euclidean space denoted by Rn.
(iii) A metric space is a space in which distance between elements is defined. This implies that an Euclidean space is a metric space. However, all metric spaces are not necessarily Euclidean. Nonetheless, some basic properties of the distance definition in the Euclidean space are present in all metric spaces. These basic properties are:
(a) |p1 - p2|≥0; |p1 - p2| = 0 if and only if p1 = p2. This is the positve definite property.
(b) |p1 - p2| = |p2 - p1|. This is the symmetric property.
(c) |p1 - p2| ≤ |p1 - p3| + |p3 - p2|. This is the triangle inequality property.

Problem 32: define a metric or distance function on a set of points X. Then state a formal definition for a metric space.
Ans: a metric or distance function on a set is a real-valued function d, whose domain and range are in X and which has the positive definite, symmetric and triangle inequality properties, for all x, y, and z in X. So:
(i) d(x,y) ≥ 0; d(x,y) = 0 if and only if x = y;
(ii) d(x,y) = d(y,x);
(iii) d(x,y) ≤ d(x,z) + d(z,y).
Consequently, a metric space, denoted as (X,d) is a nonempty set X and a metric d defined on X. The elements of X are called the points of X.

Problem 33: if the metric space (X,d) is a 3-dimensional Euclidean space what is the value of
|x - y| if x = (1,2,0) and y = (0,2,1)
Ans: √2

Problem 34 : A is the set (7,14); B is the set [7,14]. What numbers are contained in B but are not contained in A?
Ans: 7 and 14. A = {x : 7<x<14}; B = {x : 7≤x≤14}. A is an open set and B is a closed set.

Problem 35 : formalize the concept of open sets and closed sets.
Ans: consider the metric space (X,d). Let x be any point in X and r any positive real number.
The set B(x;r) = {y:d(x,y) <r} is called the open ball with center x and radius r.
For any set A in X:
x is an interior point of A if some open ball with center x is contained in A;
x is a boundary point of A if every open ball with center x contains at least one point of A and at least one point of ~A (the compliment of A);
x is an exterior point of A if some open ball with center x is contained in ~A.
The sets of all interior, boundary, and exterior points of A are called the interior, boundary and exterior of A respectively and are denoted by Int(A), Bdy(A), and Ext(A).
Consequently, X = Int(A)UBdy(A)UExt(A), that is, the union of the interior, boundary and exterior of A.
In general, a set E in a metric space is open if all points of E are interior points. E is a closed set called the closure of E if E = the union of its interior points and its boundary points, that is,
E = Int(E) U Bdy(E). E is neither open nor close if it contains only some of its boundary points.

Problem 36: what is the difference between a metric space and a topological space?
Ans: a metric space is a topological space. However, a topological space that is not metrizable (i.e.,a metric is not defined) cannot be a metric space.
Consider a nonempty set X and a collection of subsets of X, C satisfying the following conditions:
(i) the empty set and X itself are in C;
(ii) the intersection of any finite collection of sets in C is a set in C;
(iii) the union of any collection of sets in C is a set in C.
The collection C is called a topology for X and the pair (X,C) is called a topological space.

Problem 37: the triple concepts of compactness, continuity and connectedness are important concepts in a metric space. Now suppose (X, d1) and (Y, d2) are metric spaces and X is compact and connected:
(i) What is the meaning of the compactness and connectedness of X?
(ii) Is Y connected if a continuous function f maps X onto Y?
Ans: (i) let S be an index set and {Us: s∈S) a collection of open sets in X. {Us: s∈S) is an open cover of a subset E of X if E ⊂ Us∈SUs, that is, E is contained in the union of {Us: s∈S}. A finite subcollection {Us: s∈F) of sets in the open cover {Us: s∈S) is called a finite subcover of E if
E ⊂ Us∈FUs.
A subset E of a metric space X is compact if every open cover of E contains a finite subcover. So the metric space X is compact if the set X is compact.
The metric space is connected if and only if, the empty set, ∅ and the set X are the only sets in the metric space X which are both open and closed.
(ii) Yes, Y is connected.

Problem 38: state the Heine-Borel Theorem. Does the theorem hold in an arbitrary metric space?
Ans: any open cover {Us: s∈S} of a closed and bounded set E in Rn contains a finite subcover of E.
No. The Heine-Borel Theorem does not hold in an arbtrary metric space. A compact set is closed and bounded, however, a closed and bounded set is not necessarily compact.

Problem 39: is the Euclidean space Rn connected?
Ans: yes.

Problem 40:Determine the connectedness of the following:
the set A =[0,1) and the set B = (1,2]. In other words, Is A and B separate or connected?
Ans: try it.

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Peter Oye Sagay

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