Master General Topology Concepts: Interactive Learning Quizzes

Question 1

Which of the following is NOT a characteristic property of a topological space?

Accepted Answer

Order

Answer

Continuity

Answer

Connectedness

Answer

Compactness

Question 2

Let X and Y be topological spaces. A function f: X -> Y is continuous if and only if for every:

Accepted Answer

Open set in X, the image under f is an open set in Y.

Answer

Closed set in X, the image under f is a closed set in Y.

Answer

Compact set in X, the image under f is a compact set in Y.

Answer

Path-connected set in X, the image under f is a path-connected set in Y.

Question 3

Let X be a topological space and A a subset of X. The closure of A, denoted by Cl(A), is defined as:

Accepted Answer

The intersection of all closed sets containing A

Answer

The union of all open sets containing A

Answer

The set of all limit points of A

Answer

The set of all interior points of A

Question 4

Let X be a topological space. A function f: X -> R is continuous if and only if:

Accepted Answer

The preimage of every open set in R is open in X.

Answer

The preimage of every closed set in R is closed in X.

Answer

The image of every open set in X is open in R.

Answer

The image of every closed set in X is closed in R.

Question 5

Which of the following is NOT considered a homeomorphism?

Accepted Answer

Inverting the order of elements in a set

Answer

Translation of a metric space

Answer

Rotation of a circle

Answer

Projection of a plane onto a line

Question 6

Let X be a topological space and A a subset of X. The interior of A, denoted by Int(A), is defined as:

Accepted Answer

The union of all open sets contained in A

Answer

The intersection of all closed sets containing A

Answer

The set of all limit points of A

Answer

The set of all isolated points of A

Question 7

In the topological space (X, T), with A being a subset of X, what is the correct statement about the closure of A, denoted as cl(A)?

Accepted Answer

cl(A) is the smallest closed set that contains A.

Answer

cl(A) is always an open set in X.

Answer

cl(A) is the complement of the interior of A.

Answer

cl(A) is always a compact set.

Question 8

Which condition is necessary for a collection C of subsets of a set X to be a basis for a topology on X?

Accepted Answer

Every open set in X is a union of sets in C

Answer

The union of all sets in C is X

Answer

The intersection of any two sets in C is in C

Answer

Every set in C is both open and closed

Question 9

Let X be a compact topological space and f: X -> R be a continuous function. Which statement is true?

Accepted Answer

f is bounded

Answer

f is continuous and injective

Answer

f maps X onto R

Answer

f has an inverse function

Question 10

Which condition is sufficient for the product space X x Y of two topological spaces to be connected?

Accepted Answer

X is connected and Y is connected

Answer

X is compact and Y is connected

Answer

X is connected and Y is compact

Answer

X is compact and Y is compact

Question 11

Which property is NOT a characteristic of compact sets in a metric space?

Accepted Answer

Open

Answer

Closed and bounded

Answer

Totally bounded

Question 12

Which of the following is the correct definition of a topological space?

Accepted Answer

A set of points X equipped with a collection of subsets τ of X satisfying the following axioms: (1) X and the empty set are in τ. (2) The union of any number of sets in τ is also in τ. (3) The intersection of any finite number of sets in τ is also in τ.

Answer

A set of points equipped with a distance function

Answer

A set of points equipped with a closed neighbourhood system

Answer

A set of points equipped with a filter

Question 13

Let (X, τ) be a topological space. Which of the following is a necessary condition for X to be separable?

Accepted Answer

There exists a dense countable subset of X

Answer

Every open cover of X has a finite subcover

Answer

X is a compact space

Answer

X is a first-countable space

Question 14

Which property is NOT fundamental to defining a topological space?

Accepted Answer

The structure of a metric space

Answer

A set of open sets

Answer

Adherence to the three axioms of topology

Answer

A neighborhood system for each point

Question 15

For a function f: X → Y operating on a topological space X, which condition ensures its continuity at point x ∈ X?

Accepted Answer

The preimage of every open set in Y is open in X.

Answer

The preimage of every closed set in Y is closed in X.

Answer

The image of every open set in X is open in Y.

Answer

The image of every closed set in X is closed in Y.

Question 16

Which set of intervals forms a basis for the topology on the real line?

Accepted Answer

All open intervals

Answer

All closed intervals

Answer

All half-open intervals

Answer

All singleton sets

Question 17

Which of the following topological spaces is an example of a connected space?

Accepted Answer

The set of real numbers with the usual topology

Answer

The set of rational numbers with the usual topology

Answer

The set of integers with the discrete topology

Answer

The set of points in the plane that lie on the x-axis

Question 18

Which of the following subsets of the real line is compact?

Accepted Answer

[0, 1]

Answer

(0, 1)

Answer

[-∞, ∞]

Answer

The set of all rational numbers

Question 19

Which of the following functions is a homeomorphism between the unit circle and the square [-1, 1] x [-1, 1]?

Accepted Answer

f(x, y) = (x/√2, y/√2)

Answer

f(x, y) = (x, y)

Answer

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

Answer

f(x, y) = (x/2, y/2)

Question 20

Which of the following sets of subsets of the real line forms a basis for the standard topology?

Accepted Answer

The set of all open intervals

Answer

The set of all closed intervals

Answer

The set of all half-open intervals

Answer

The set of all finite intervals

Question 21

Which property is NOT essential for a topological space?

Accepted Answer

It exhibits connected components.

Answer

It is a set with a defined topology.

Answer

It has a notion of open and closed sets.

Answer

Its open sets satisfy the conditions of arbitrary unions and finite intersections.

Question 22

Which topological space exhibits connectedness?

Accepted Answer

The set of all real numbers.

Answer

The set of all rational numbers.

Answer

The set of all integers.

Answer

The set of all complex numbers.

Question 23

Which property is equivalent to a topological space being Hausdorff?

Accepted Answer

Any two distinct points have disjoint neighborhoods.

Answer

All points in the space are closed.

Answer

All subsets are compact.

Answer

All open covers have finite subcovers.

Question 24

In a topological space X, what is a basis for the topology?

Accepted Answer

A collection of open sets covering X.

Answer

A collection of closed sets covering X.

Answer

A collection of sets that are both open and closed in X.

Answer

A collection of sets that are neither open nor closed in X.

Question 25

Which property is NOT preserved under homeomorphisms?

Accepted Answer

Cardinality

Answer

Compactness

Answer

Connectedness

Answer

Hausdorff property

Question 26

In a topological space X, what is the defining characteristic of the closure of a set A?

Accepted Answer

It is the smallest closed set that contains A.

Answer

It is the largest open set that contains A.

Answer

It is the set of all limit points of A.

Answer

It is the complement of the interior of A.

Question 27

Let (X, T) be a topological space and A ⊆ X. Which of the following is equivalent to A being a dense subset of X?

Accepted Answer

Every non-empty open subset of X intersects A

Answer

The closure of A is equal to X

Answer

A is a compact subspace of X

Answer

A is a connected subspace of X

Question 28

State the set that is NOT a topological space.

Accepted Answer

Set of subsets of a finite set

Answer

Set of real numbers with standard topology

Answer

Set of rational numbers with discrete topology

Answer

Set of natural numbers with co-finite topology

Question 29

Which of the following is a necessary condition for a topological space (X, τ) to be compact?

Accepted Answer

Every open cover of X has a finite subcover.

Answer

Every infinite subset of X has a limit point.

Answer

Every continuous function from X to R is uniformly continuous.

Answer

Every sequence in X has a convergent subsequence.

Question 30

In a Hausdorff space (X, τ), which of the following properties holds?

Accepted Answer

Every pair of distinct points in X can be separated by disjoint open sets.

Answer

Every open set in τ is compact.

Answer

Every closed set in τ is connected.

Answer

Every continuous function from X to R is differentiable.

Question 31

Which of the following is a necessary condition for a metric space (X, d) to be complete?

Accepted Answer

Every Cauchy sequence in X converges to a point in X.

Answer

Every bounded sequence in X has a convergent subsequence.

Answer

Every closed ball in X is compact.

Answer

Every open ball in X is connected.

Question 32

For a function f: X -> Y between topological spaces X and Y, which condition guarantees that f is continuous?

Accepted Answer

The preimage of every open set in Y is open in X

Answer

f is injective (one-to-one)

Answer

f is surjective (onto)

Answer

The image of every closed set in X is closed in Y

Question 33

A topological space X is connected if:

Accepted Answer

It cannot be separated into two disjoint non-empty open sets

Answer

It can be separated into two disjoint non-empty closed sets

Answer

It is compact

Answer

It is Hausdorff

Question 34

Which of the following is equivalent to the Heine-Borel theorem?

Accepted Answer

Every open cover of a compact space has a finite subcover

Answer

Every closed cover of a compact space has a finite subcover

Answer

Every open cover of a connected space has a finite subcover

Answer

Every closed cover of a connected space has a finite subcover

Question 35

Which of the following properties is NOT characteristic of a topological space?

Accepted Answer

Finite cardinality

Answer

Open sets

Answer

Continuity

Answer

Closed sets

Question 36

In a topological space (X, τ), which condition is equivalent to the continuity of a function f: X → Y?

Accepted Answer

The preimage of every open set in Y is open in X.

Answer

The image of every open set in X is open in Y.

Answer

The preimage of every closed set in Y is closed in X.

Question 37

Which of the following is NOT a connected space?

Accepted Answer

The disjoint union of two open intervals in the real line

Answer

A solid sphere

Answer

A circle

Question 38

Which of the following sets is compact in the real line topology?

Accepted Answer

The closed unit interval [0, 1]

Answer

The set of all real numbers

Answer

The open unit interval (0, 1)

Answer

The set of all rational numbers between 0 and 1

Question 39

Which collection of sets forms a basis for the topology τ on a topological space (X, τ)?

Accepted Answer

The collection of all open sets in X

Answer

The collection of all finite subsets of X

Answer

The collection of all closed sets in X

Answer

The collection of all compact sets in X

Question 40

Which of the following topologies is a product topology?

Accepted Answer

The topology on the Cartesian product of two topological spaces

Answer

The topology on the power set of a topological space

Answer

The topology on the disjoint union of two topological spaces

Question 41

Which of the following functions is a homeomorphism?

Accepted Answer

A bijective continuous function with a continuous inverse

Answer

A continuous function

Answer

A bijective function

Question 42

A topological space is defined as:

Accepted Answer

A pair (X, τ) where X is a set and τ is a collection of subsets of X called open sets that satisfy certain axioms.

Answer

A set of points.

Question 43

A connected topological space is defined such that:

Accepted Answer

It cannot be expressed as the union of two disjoint open sets.

Answer

It is always Hausdorff.

Answer

It is always compact.

Question 44

A topological space is considered compact if:

Accepted Answer

Every open cover of the space has a finite subcover.

Answer

Every closed cover of the space has a finite subcover.

Answer

The space is bounded and closed.

Question 45

Which statement is a necessary and sufficient condition for a topological space to be Hausdorff?

Accepted Answer

If x ≠ y, then there exist disjoint open sets containing x and y.

Answer

The space is connected.

Answer

The space is compact.

Question 46

The product topology on X × Y is defined as:

Accepted Answer

The smallest topology on X × Y containing all sets of the form U × V, where U is open in X and V is open in Y.

Answer

The topology generated by the metric d((x1, y1), (x2, y2)) = |x1 - x2| + |y1 - y2|

Question 47

Topology finds applications in computer science. Which of the following is an example?

Accepted Answer

Identifying connected components of a graph.

Answer

Solving differential equations.

Answer

Finding roots of polynomial equations.

Answer

Calculating integrals.