Master Metric Spaces: A Comprehensive Guide for Real Analysis Students

Question 1

In a metric space (X, d), a set E is open if every point in E has a neighborhood of radius r such that the open ball centered at the point with radius r is contained in E.

Accepted Answer

neighborhood

Answer

boundary

Answer

closure

Answer

interior

Question 2

Which of the following is NOT a property of closed sets in a metric space?

Accepted Answer

They are always bounded.

Answer

They contain all their limit points.

Answer

They are complements of open sets.

Answer

They are compact if the metric space is complete.

Question 3

A sequence {x_n} in a metric space (X, d) is called a Cauchy sequence if for every positive real number ε there exists an integer N such that d(x_m, x_n) < ε whenever m, n > N.

Accepted Answer

positive real number

Answer

neighborhood

Answer

open set

Answer

closed set

Question 4

Define a metric space and explain its significance in mathematical analysis.

Accepted Answer

A metric space (X, d) consists of a set X and a function d: X x X → R, called the distance function or metric, that satisfies the following properties for all x, y, z in X:

Answer

A metric space is a collection of points with defined distances between them.

Answer

A metric space is a space where the distance between any two points is always positive.

Answer

A metric space is a set of points where the distance between any two points is unique.

Question 5

In the metric space (R, usual distance function), which of the following sets is closed?

Accepted Answer

The set of all numbers greater than or equal to 1

Answer

The set of all numbers less than 1

Answer

The set of all numbers between 0 and 1

Answer

The set of all rational numbers

Question 6

Which of the following is a homeomorphism between the metric spaces (R, usual distance function) and (R^2, usual distance function)?

Accepted Answer

f(x) = (x, 0)

Answer

f(x) = (x, x^2)

Answer

f(x) = (x, |x|)

Answer

f(x) = (sin(x), cos(x))

Question 7

How does the topology of a metric space influence the convergence of sequences within that space?

Accepted Answer

The topology determines whether sequences converge or diverge.

Answer

The topology has no bearing on sequence convergence.

Answer

All sequences in a metric space converge.

Answer

The topology of a metric space is solely concerned with open and closed sets.

Question 8

Which of the following is not a property of a metric space?

Accepted Answer

The distance between any two points can be negative.

Answer

The distance between two points is symmetric.

Answer

The distance between any two points is non-negative.

Answer

The distance between any two points is unique.

Question 9

In the metric space (X, d), which set is open?

Accepted Answer

The set of all points in X within a distance of 1 from a fixed point p.

Answer

The set of all points in X within a distance of 0 from a fixed point p.

Answer

The set of all points in X outside a distance of 1 from a fixed point p.

Question 10

Which of the following is the definition of an open set in a metric space?

Accepted Answer

A set of points that can be separated from its complement by an open ball

Answer

A set of points that is dense in the space

Answer

A set of points that contains its boundary

Question 11

Which of the following is a necessary and sufficient condition for a metric space to be complete?

Accepted Answer

Every Cauchy sequence in the space converges to a point in the space

Answer

The space is compact

Answer

The space is bounded

Question 12

Which of the following is a typical application of metric spaces?

Accepted Answer

Modeling the space of all possible states of a physical system

Answer

Solving differential equations

Answer

Finding the roots of polynomials

Question 13

Consider the metric space (R2, d), where d((x1, y1), (x2, y2)) = sqrt((x1 - x2)2 + (y1 - y2)2). Which of the following sets is open?

Accepted Answer

{(x, y) | x2 + y2 < 1}

Answer

{(x, y) | x2 + y2 <= 1}

Answer

{(x, y) | x > 0}

Answer

{(x, y) | x = y}

Question 14

Consider the metric space (C, d), where d(z1, z2) = |z1 - z2| for complex numbers z1 and z2. Which of the following sets is complete?

Accepted Answer

C

Answer

Q

Answer

Z

Answer

R

Question 15

Which of the following sets, equipped with the Euclidean distance, forms a metric space?

Accepted Answer

The set of real numbers

Answer

The set of integers

Answer

The set of rational numbers

Answer

The set of all continuous functions on the interval [0, 1]

Question 16

Given a metric space (X, d), which of the following statements is true?

Accepted Answer

If X is compact and d is continuous, then (X, d) is complete.

Answer

If X is complete and d is continuous, then X is compact.

Answer

If X is compact, then (X, d) is complete.

Question 17

In the metric space (R, |·|), which of the following is an example of a Cauchy sequence?

Accepted Answer

{1/n : n ∈ N}

Answer

{(-1)^n : n ∈ N}

Answer

{n : n ∈ N}

Answer

{sin(n) : n ∈ N}

Question 18

Let (X, d) be a metric space. Which of the following sets is a closed subset of X?

Accepted Answer

The set of all x ∈ X such that d(x, a) ≤ r, where a ∈ X and r > 0.

Answer

The set of all x ∈ X such that d(x, a) = r, where a ∈ X and r > 0.

Answer

The set of all x ∈ X such that d(x, a) < r, where a ∈ X and r > 0.

Question 19

Let (X, d) be a metric space. Which of the following sets is an open subset of X?

Accepted Answer

The set of all x ∈ X such that d(x, a) < r, where a ∈ X and r > 0.

Answer

The set of all x ∈ X such that d(x, a) ≤ r, where a ∈ X and r > 0.

Answer

The set of all x ∈ X such that d(x, a) = r, where a ∈ X and r > 0.

Question 20

Let (X, d) be a metric space. Which of the following is a complete metric space?

Accepted Answer

The set of all real numbers in X with the usual metric.

Answer

The set of all integers in X with the usual metric.

Answer

The set of all rational numbers in X with the usual metric.

Question 21

Which of the following is **not** a property that must be satisfied by a distance function (metric) in a metric space?

Accepted Answer

Boundedness

Answer

Completeness

Answer

Non-negativity

Answer

Triangle inequality

Question 22

Consider the set of real numbers with the usual metric (absolute value). Which of the following intervals is an open set?

Accepted Answer

(0, 1)

Answer

[0, 1]

Answer

(0, 1]

Answer

[0, 1)

Question 23

Which of the following sets, equipped with the usual metric, is a complete metric space?

Accepted Answer

The set of real numbers

Answer

The set of integers

Answer

The set of finite subsets of natural numbers

Answer

The set of rational numbers

Question 24

In a metric space (X, d), what is the radius of the open ball centered at a point x ∈ X with radius 0.5?

Accepted Answer

0.5

Answer

2

Answer

1

Answer

0.25

Question 25

Let (X, d) be a metric space. Which of the following conditions **guarantees** that (X, d) is a discrete metric space?

Accepted Answer

For all x, y ∈ X, d(x, y) = 0 or 1

Answer

For all x, y ∈ X, d(x, y) ≤ 1

Answer

For all x, y ∈ X, d(x, y) ≥ 0

Question 26

Which of the following functions defines a valid metric on the set of real numbers?

Accepted Answer

d(x, y) = |x - y|

Answer

d(x, y) = x + y

Answer

d(x, y) = xy

Answer

d(x, y) = 1 / (x - y)

Question 27

Let (X, d) be a metric space and let f: X → Y be a function between metric spaces. Which of the following statements is the **formal definition** of continuity for f?

Accepted Answer

For all ε > 0, there exists δ > 0 such that for all x, y ∈ X, d(x, y) < δ implies d(f(x), f(y)) < ε

Answer

For all x, y ∈ X, d(x, y) < d(f(x), f(y))

Answer

For all x, y ∈ X, d(f(x), f(y)) ≤ d(x, y)

Question 28

Which of the following statements does **NOT** hold true for a metric *d* defined on a set *X*?

Accepted Answer

d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ X

Answer

d(x, y) = 0 if and only if x = y

Answer

d(x, y) ≥ 0 for all x, y ∈ X

Answer

d(x, y) = d(y, x) for all x, y ∈ X

Question 29

In a metric space (X, d), the open ball B(x, r) centered at *x* with radius *r* is always:

Accepted Answer

an open set in (X, d)

Answer

neither open nor closed in (X, d)

Answer

a closed set in (X, d)

Answer

both open and closed in (X, d)

Question 30

In the metric space (ℝ, d) where d(x, y) = |x - y|, which of the following sets is **CLOSED**?

Accepted Answer

{x ∈ ℝ | x ≤ 1}

Answer

{x ∈ ℝ | x ≠ 0}

Answer

{x ∈ ℝ | x < 1}

Answer

{x ∈ ℝ | 0 < x < 1}

Question 31

A sequence (x_n) in a metric space (X, d) is considered a Cauchy sequence if:

Accepted Answer

for any ε > 0, there exists an N ∈ ℕ such that d(x_m, x_n) < ε for all m, n ≥ N

Answer

for any ε > 0, there exists an N ∈ ℕ such that d(x_m, x_n) = ε for all m, n ≥ N

Answer

for any ε > 0, there exists an N ∈ ℕ such that d(x_m, x_n) > ε for all m, n ≥ N

Question 32

A metric space (X, d) is deemed complete if:

Accepted Answer

every Cauchy sequence in (X, d) converges to a limit within X

Answer

every sequence in (X, d) converges to a limit within X

Answer

every bounded sequence in (X, d) converges to a limit within X

Question 33

Which of the following metric spaces is **NOT** complete?

Accepted Answer

(ℚ, d) where d(x, y) = |x - y|

Answer

([0, 1], d) where d(x, y) = |x - y|

Answer

(ℝ, d) where d(x, y) = |x - y|

Answer

(C[0, 1], d) where d(f, g) = max{|f(x) - g(x)| : x ∈ [0, 1]}

Question 34

The space of continuous functions on the interval [0, 1] with the sup norm metric, denoted as (C[0, 1], ||.||_∞), is:

Accepted Answer

complete

Answer

finite dimensional

Answer

bounded

Answer

not complete

Question 35

In a metric space (X, d) and a subset A ⊆ X, the closure of A, denoted as cl(A), is:

Accepted Answer

the smallest closed set that contains A

Answer

the largest open set that contains A

Answer

the set of all interior points of A

Answer

the set of all limit points of A

Question 36

In the metric space (ℝ, d) where d(x, y) = |x - y|, which of the following sets is **dense** in ℝ?

Accepted Answer

ℚ (the set of rational numbers)

Answer

{x ∈ ℝ | x ≤ 1}

Answer

ℤ (the set of integers)

Answer

{x ∈ ℝ | x > 0}

Question 37

Which of the following is **NOT** a property of a continuous function f: (X, d) → (Y, ρ) between metric spaces?

Accepted Answer

f(x) is a limit point of Y for every limit point x of X

Answer

f preserves convergent sequences: if (x_n) converges to x in (X, d), then (f(x_n)) converges to f(x) in (Y, ρ)

Answer

f preserves open sets: the pre-image of any open set in (Y, ρ) under f is an open set in (X, d)

Question 38

Which of the following is NOT a property of a metric *d* on a set *X*?

Accepted Answer

d(x, y) < 0 for some x, y in X

Answer

d(x, x) = 0 for all x in X

Answer

d(x, y) = d(y, x) for all x, y in X

Question 39

In a metric space *(X, d)*, which of the following sets is always considered open?

Accepted Answer

{x ∈ X : d(x, a) < r} for some a ∈ X and r > 0

Answer

{x ∈ X : d(x, a) ≤ r} for some a ∈ X and r > 0

Answer

{x ∈ X : d(x, a) = r} for some a ∈ X and r > 0

Question 40

Consider the metric space (ℝ, |·|).  Which of these sets is closed?

Accepted Answer

[0, 1]

Answer

(0, 1]

Answer

[0, 1)

Answer

(0, 1)

Question 41

In a metric space *(X, d)*, a sequence *(x_n)* is called a Cauchy sequence if...

Accepted Answer

For any ε > 0, there exists a natural number *N* such that *d(x_m, x_n) < ε* for all *m, n ≥ N*

Answer

The sequence *(x_n)* converges to a limit in *X*.

Answer

For any ε > 0, there exists a natural number *N* such that *d(x_n, x) < ε* for all *n ≥ N*, for some *x ∈ X*

Question 42

A metric space *(X, d)* is called complete if...

Accepted Answer

Every Cauchy sequence in *X* converges to a limit in *X*

Answer

Every bounded sequence in *X* converges to a limit in *X*

Answer

Every sequence in *X* converges to a limit in *X*

Question 43

Which of the following is NOT a complete metric space?

Accepted Answer

(0, 1) with the usual metric

Answer

ℝ with the usual metric

Answer

C[0, 1] (space of continuous functions on [0, 1]) with the uniform metric

Answer

ℝ² with the Euclidean metric

Question 44

Let *(X, d)* be a metric space and *A ⊆ X*.  The closure of *A*, denoted by *cl(A)*, is...

Accepted Answer

The smallest closed set containing *A*

Answer

The set of all points in *A* that are not limit points of *A*

Answer

The set of all limit points of *A*

Answer

The largest open set contained in *A*

Question 45

Let *(X, d)* be a metric space.  Which of the following statements is true about closed sets?

Accepted Answer

If *A* and *B* are closed sets, then *A ∩ B* is also closed

Answer

If *A* and *B* are open sets, then *A ∪ B* is also open

Answer

If *A* is closed, then *Aᶜ* (complement of *A*) is also closed

Question 46

Let *(X, d)* be a metric space.  A point *x ∈ X* is called an isolated point of a set *A ⊆ X* if...

Accepted Answer

There exists an ε > 0 such that the open ball *B(x, ε)* intersects *A* only at *x*

Answer

*x* is an interior point of *A*

Answer

*x* is a limit point of *A*

Question 47

Consider the metric space (ℝ, |·|) and the sequence *(x_n)* = (1/n). Which of the following statements is true about this sequence?

Accepted Answer

The sequence *(x_n)* converges to 0

Answer

The sequence *(x_n)* is a Cauchy sequence but does not converge

Answer

The sequence *(x_n)* converges to 1

Answer

The sequence *(x_n)* is not a Cauchy sequence