# Topology : Topology

## Example Questions

### Example Question #1 : Introduction

What is the simplex in the zero dimension?

Object

Vertex

Edge

Line

Face

Vertex

Explanation:

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 #2 : Introduction

What is the simplex in the second dimension?

Face

Vertex

Edge

Line

Space

Face

Explanation:

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 #3 : Introduction

What is the simplex in the first dimension?

Point

Object

Vertex

Edge

Face

Edge

Explanation:

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.

Explanation:

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.

Explanation:

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.

None of the answers are correct.

Explanation:

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?

None of the answers are correct.

Explanation:

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.

A line doesn't have an Euler Characteristic.

Explanation:

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?

Explanation:

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?

A point doesn't have an Euler Characteristic.