### All Topology Resources

## Example Questions

### Example Question #1 : Betti Numbers

What is the simplex in the zero dimension?

**Possible Answers:**

Edge

Vertex

Face

Line

Object

**Correct answer:**

Vertex

In topology, Betti numbers represent the various dimensions of a topological space in regards to the number of holes that are present. There are four different simplices depending on the dimension: zero, one, two, three.

These are also referred to as:

Zero simplex refers to a point or vertex.

One simplex refers to a line or edge (which is connected by two points)

Two simplex refers to a face (which is connected by three lines)

Three simplex refers to the 3D object created by two faces.

Therefore the simplex in the zero dimension is known as a vertex.

### Example Question #1 : Topology

What is the simplex in the second dimension?

**Possible Answers:**

Edge

Vertex

Face

Line

Space

**Correct answer:**

Face

In topology, Betti numbers represent the various dimensions of a topological space in regards to the number of holes that are present. There are four different simplices depending on the dimension: zero, one, two, three.

These are also referred to as:

Zero simplex refers to a point or vertex.

One simplex refers to a line or edge (which is connected by two points)

Two simplex refers to a face (which is connected by three lines)

Three simplex refers to the 3D object created by two faces.

Therefore the simplex in the second dimension is known as a face. What is the simplex in the zero dimension?

### Example Question #2 : Topology

What is the simplex in the first dimension?

**Possible Answers:**

Object

Vertex

Face

Edge

Point

**Correct answer:**

Edge

In topology, Betti numbers represent the various dimensions of a topological space in regards to the number of holes that are present. There are four different simplices depending on the dimension: zero, one, two, three.

These are also referred to as:

Zero simplex refers to a point or vertex.

One simplex refers to a line or edge (which is connected by two points)

Two simplex refers to a face (which is connected by three lines)

Three simplex refers to the 3D object created by two faces.

Therefore the simplex in the first dimension is known as a edge.

### Example Question #4 : Betti Numbers

Calculate the Euler Characteristic given the following Betti numbers.

**Possible Answers:**

**Correct answer:**

The Euler Characteristic is calculated by the following equation.

This is also written as,

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

In this particular question the Betti numbers are known therefore, substitute them into the formula to calculate the Euler Characteristic and solve.

### Example Question #5 : Betti Numbers

Calculate the Euler Characteristic given the following Betti numbers.

**Possible Answers:**

**Correct answer:**

The Euler Characteristic is calculated by the following equation.

This is also written as,

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

In this particular question the Betti numbers are known therefore, substitute them into the formula to calculate the Euler Characteristic and solve.

### Example Question #1 : Euler Characteristic

Calculate the Euler Characteristic given the following Betti numbers.

**Possible Answers:**

None of the answers are correct.

**Correct answer:**

The Euler Characteristic is calculated by the following equation.

This is also written as,

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

In this particular question the Betti numbers are known therefore, substitute them into the formula to calculate the Euler Characteristic and solve.

### Example Question #2 : Euler Characteristic

What is the Euler Characteristic associated with a 2-dimensional square?

**Possible Answers:**

None of the answers are correct.

**Correct answer:**

The Euler Characteristic is calculated by the following equation.

This is also written as,

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

In this particular question the Betti numbers can be calculated therefore,

now substitute them into the formula to calculate the Euler Characteristic and solve.

### Example Question #3 : Euler Characteristic

What is the Euler Characteristic for a line.

**Possible Answers:**

A line doesn't have an Euler Characteristic.

**Correct answer:**

The Euler Characteristic is calculated by the following equation.

This is also written as,

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

In this particular question the Betti numbers can be calculated therefore,

now substitute them into the formula to calculate the Euler Characteristic and solve.

### Example Question #4 : Euler Characteristic

What is the Euler Characteristic associated with a 3-dimensional cube?

**Possible Answers:**

**Correct answer:**

The Euler Characteristic is calculated by the following equation.

This is also written as,

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

In this particular question the Betti numbers can be calculated therefore,

now substitute them into the formula to calculate the Euler Characteristic and solve.

### Example Question #5 : Euler Characteristic

What is the Euler Characteristic for a point?

**Possible Answers:**

A point doesn't have an Euler Characteristic.

**Correct answer:**

The Euler Characteristic is calculated by the following equation.

This is also written as,

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

In this particular question the Betti numbers can be calculated therefore,

now substitute them into the formula to calculate the Euler Characteristic and solve.