6 Metric spaces

Question 1

Metric, metric space, distance

Answer 1

Let X be a non-empty set and d a real-valued function defined on X x X such that for a, b e X:
i) d(a,b) >= 0 and d(a,b) = 0 if and only if a = b;
ii) d(a,b) = d(b,a); and
iii) d(a,c) <= d(a,b) + d(b,c), [the triangle inequality] for all a,b and c in X.
Then d is said to be a metric on X, (X,d) is called a metric space and d(a,b) is referred to as the distance between a and b.

Question 2

Open ball

Answer 2

Let (X, d) be a metric space and r any positive real number. Then the open ball about a e X of radius r is the B_r(a) = {x: x e X and d(a,x) < r}

Question 3

Equivalent metrics

Answer 3

Metrics on a set X are said to be equivalent if they induce the same topology on X.

Question 4

Hausdorff space (T_2 space)

Answer 4

A topological space is said to be a Hausdorff space if for each pair of distinct points a and b in X, there exist open sets U and V such that a e U, b e V, and U n B = /O.

Question 5

Metrizable space

Answer 5

A space (X, T) is said to be metrizable if there exists a metric d on the set X with the property that T is the topology induced by d.

Question 6

To converge to / convergent

Answer 6

Let (X, d) be a metric space and x1, …, xn, … a sequence of points in X. Then the sequence is said to converge to x e X if given any epsilon > 0 there exists an integer n0 such that for all n >= n0, d(x, xn) < epsilon. This is denoted by xn -> x. The sequence y1, y2, …, yn, … of points in (X,d) is said to be convergent if there exists a point y e X such that yn -> y.

Question 7

Cauchy sequence

Answer 7

A sequence x1, x2, …, xn, .. of points in a metric space (X, d) is said to be a Cauchy sequence if given any real number epsilon > 0, there exists a positive integer n0, such that for all integers m >= n0 and n >= n0, d(xm, xn) < epsilon.

Question 8

Completeness

Answer 8

A metric space (X, d) is said to be complete if every Cauchy sequence in (X, d) converges to a point in (X, d).

Question 9

Subsequence

Answer 9

If {xn} is any sequence, then the sequence x_n1, x_n2, … is said to be a subsequence if n1 < n2 < n3 < …

Question 10

Increasing sequence / decreasing sequence

Answer 10

Let {x_n} be a sequence in R. Then it is said to be an increasing sequence if x_n <= x_n+1, for all n e N. It is said to be decreasing sequence if x_n >= x_n+1 for all n e N.

Question 11

Monotonic sequence

Answer 11

A sequence that is either increasing or decreasing.

Question 12

Peak point

Answer 12

Let {x_n} be a sequence in R. Then n_0 e N is said to be a peak point if x_n <= x_n0, for every n>= n0

Question 13

Completely metrizable space

Answer 13

A topological space (X, T) is said to be completely metrizable if there exists a metric d on X such that T is the topology on X determined by d and (X, d) is a complete metric space.

Question 14

Separable space

Answer 14

A topological space is said to be separable if it has a countable dense subset.

Question 15

Polish space

Answer 15

A topological space (X,T) is said to be a Polish space if it is separable and completely metrizable,

Question 16

Souslin space (Suslin space)

Answer 16

A topological space (X, T) is said to be a Souslin space if it is a Hausdorff and a continous image of a Polish space.

Question 17

Analytic set

Answer 17

If A is a subset of a topological space (Y, T1) such that with the induced topology T2, the space (A, T2) is a Souslin space, then A is said to be an an analytic set in (Y, T1).

Question 18

Isometric spaces, Isometry

Answer 18

Let (X, d) and (Y, d1) be metric spaces. Then (X, d) is said to be isometric to (Y, d1) if there exists a surjective mapping f: X -> Y such that for all x1 and x2 in X, d(x1,x2) = d1(f(x1), f(x2)). Such a mapping f is said to be an isometry.

Question 19

Isometric embedding

Answer 19

Let (X, d) and (Y,d1) be metric spaces and f a mapping of X into Y. Let Z = f(X), and d2 be the metric induced on Z by d1. If f: (X,d) -> (Z, d2) is an isometry, then f is said to be an isometric embedding of (X, d) in (Y, d1).

Question 20

Completion

Answer 20

Let (X, d) and (Y, d1) be metric spaces and f a mapping of X into Y. If (Y, d1) is a complete metric space, f: (X,d) -> (Y, d1) is an isometric embedding and f(X) is a dense subset of Y in the associated topological space, then (Y, d1) is said to be a completion of (X,d).

Question 21

Banach space

Answer 21

Let (N, || ||) be a normed vector space and d the associated metric on the set N. Then (N, || ||) is said to be a Banach space if (N, d) is a complete metric space.

Question 22

Fixed point

Answer 22

Let f be a mapping of a set X into itself. Then a point x e X is said to be a fixed point of f if f(x) = x.

Question 23

Contraction mapping

Answer 23

Let (X, d) be a metric space and f a mapping of X into itself. Then f is said to be a contraction mapping if there exists an r e (0,1) such that
d(f(x1),f(x2)) < r.d(x1, x2) for all x1 and x2 e X.

Question 24

Interior, interior point

Answer 24

Let (X, T) be any topological space and A any subset of X. The largest open set contained in A is called the interior of A and is denoted by Int(A). Each point x e Int(A) is called an interior point of A.

Question 25

Exterior, exterior point

Answer 25

Let (X, T) be any topological space and A any subset of X. The set Int(X\A), that is the interior of the complement of A, is denoted by Ext(A), and is called the exterior of A and each point in Ext(A) is called an exterior point of A.

Question 26

Boundary, boundary point

Answer 26

Let (X, T) be any topological space and A any subset of X. The set A^- \ Int(A) is called the boundary of A. Each point in the boundary of A is called a boundary point of A.

Question 27

Nowhere dense

Answer 27

A subset A of a topological space (X, T) is said to be nowhere dense if the set A^- has empty interior.

Question 28

Baire space

Answer 28

A topological space (X, d) is said to be a Baire space if for every sequence {X_n} of open dense subsets of X, the set Intersection(X_n) is also dense in X.

Question 29

First category (meager), second category

Answer 29

Let Y be a subset of a topological space (X, T). If Y is a union of a countable number of nowhere dense subsets of X, then Y is said to be a set of the first category or meager in (X, T). If Y is not first category, it is said to be a set of the second category in (X, T).

Question 30

Convex set

Answer 30

Let S be a subset of a real vector space V. The set S is said to be convex if for each x,y e S and every real number 0 < L < 1, the point Lx + (1 - L)y is in S.