Call Now to Set Up Tutoring:

(888) 888-0446

Topology : Introduction

Study concepts, example questions & explanations for Topology

Create An Account

All Topology Resources

5 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Betti Numbers

What is the simplex in the zero dimension?

Possible Answers:

Line

Object

Vertex

Edge

Face

Correct answer:

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.

Report an Error

Example Question #2 : Betti Numbers

What is the simplex in the second dimension?

Possible Answers:

Face

Vertex

Line

Space

Edge

Correct answer:

Face

Explanation:

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?

Report an Error

Example Question #3 : Betti Numbers

What is the simplex in the first dimension?

Possible Answers:

Object

Point

Edge

Face

Vertex

Correct answer:

Edge

Explanation:

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.

Report an Error

Example Question #2 : Betti Numbers

Calculate the Euler Characteristic given the following Betti numbers.

$\\\beta_0=30 \\\beta_1=32 \\\beta_2=1$

Possible Answers:

$\chi=0$

$\chi=-2$

$\chi=-1$

$\chi=3$

$\chi=1$

Correct answer:

$\chi=-1$

Explanation:

The Euler Characteristic is calculated by the following equation.

$\text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}$

This is also written as,

$\chi=v-e+f$

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

$\\\beta_0=v \\\beta_1=e \\\beta_2=f$

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

$\\\beta_0=10 \\\beta_1=12 \\\beta_2=20$

$\\\chi=30-32+1 \\\chi=-2+1 \\\chi=-1$

Report an Error

Example Question #3 : Betti Numbers

Calculate the Euler Characteristic given the following Betti numbers.

$\\\beta_0=7 \\\beta_1=12\\\beta_2=6$

Possible Answers:

$\chi=2$

$\chi=0$

$\chi=-2$

$\chi=-1$

$\chi=1$

Correct answer:

$\chi=1$

Explanation:

The Euler Characteristic is calculated by the following equation.

$\text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}$

This is also written as,

$\chi=v-e+f$

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

$\\\beta_0=v \\\beta_1=e \\\beta_2=f$

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

$\\\beta_0=7 \\\beta_1=12\\\beta_2=6$

$\\\chi=7-12+6 \\\chi=-5+6 \\\chi=1$

Report an Error

Example Question #1 : Euler Characteristic

Calculate the Euler Characteristic given the following Betti numbers.

$\\\beta_0=4 \\\beta_1=5 \\\beta_2=2$

Possible Answers:

$\chi=1$

$\chi=2$

None of the answers are correct.

$\chi=-1$

$\chi =0$

Correct answer:

$\chi=1$

Explanation:

The Euler Characteristic is calculated by the following equation.

$\text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}$

This is also written as,

$\chi=v-e+f$

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

$\\\beta_0=v \\\beta_1=e \\\beta_2=f$

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

$\\\beta_0=4 \\\beta_1=5 \\\beta_2=2$

$\\\chi=4-5+2 \\\chi=-1+2 \\\chi=1$

Report an Error

Example Question #2 : Euler Characteristic

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

Possible Answers:

None of the answers are correct.

$\chi=1$

$\chi=-1$

$\chi=2$

$\chi=0$

Correct answer:

$\chi=1$

Explanation:

The Euler Characteristic is calculated by the following equation.

$\text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}$

This is also written as,

$\chi=v-e+f$

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

$\\\beta_0=v \\\beta_1=e \\\beta_2=f$

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

$\\\beta_0=4 \\\beta_1=4 \\\beta_2=1$

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

$\\\chi=4-4+1 \\\chi=1$

Report an Error

Example Question #3 : Euler Characteristic

What is the Euler Characteristic for a line.

Possible Answers:

$\chi=0$

$\chi=2$

$\chi=-1$

$\chi=1$

A line doesn't have an Euler Characteristic.

Correct answer:

$\chi=1$

Explanation:

The Euler Characteristic is calculated by the following equation.

$\text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}$

This is also written as,

$\chi=v-e+f$

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

$\\\beta_0=v \\\beta_1=e \\\beta_2=f$

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

$\\\beta_0=2 \\\beta_1=1 \\\beta_2=0$

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

$\\\chi=2-1+0 \\\chi=1$

Report an Error

Example Question #3 : Introduction

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

Possible Answers:

$\chi=1$

$\chi=4$

$\chi=0$

$\chi=2$

$\chi=3$

Correct answer:

$\chi=2$

Explanation:

The Euler Characteristic is calculated by the following equation.

$\text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}$

This is also written as,

$\chi=v-e+f$

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

$\\\beta_0=v \\\beta_1=e \\\beta_2=f$

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

$\\\beta_0=8 \\\beta_1=12 \\\beta_2=6$

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

$\\\chi=8-12+6 \\\chi=2$

Report an Error

Example Question #4 : Euler Characteristic

What is the Euler Characteristic for a point?

Possible Answers:

$\chi=-1$

$\chi=1$

$\chi=2$

$\chi=0$

A point doesn't have an Euler Characteristic.

Correct answer:

$\chi=1$

Explanation:

The Euler Characteristic is calculated by the following equation.

$\text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}$

This is also written as,

$\chi=v-e+f$

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

$\\\beta_0=v \\\beta_1=e \\\beta_2=f$

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

$\\\beta_0=1 \\\beta_1=0 \\\beta_2=0$

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

$\\\chi=1-0+0 \\\chi=1$

Report an Error

View Tutors

James
Certified Tutor

St. Mary of the Plains College, Bachelors, Mathematics. Baker University, Masters, Business Administration and Management.

View Tutors

Constance
Certified Tutor

Loyola University-Chicago, Bachelors, Psychology. Sam Houston State University, Masters, Counseling Psychology.

View Tutors

Anne
Certified Tutor

University of Maryland-Baltimore, Doctor of Science, Accounting.

All Topology Resources

5 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Physics Tutors in Phoenix, Computer Science Tutors in Denver, LSAT Tutors in Chicago, Algebra Tutors in Philadelphia, Reading Tutors in Atlanta, GMAT Tutors in Philadelphia, Chemistry Tutors in Seattle, Math Tutors in Dallas Fort Worth, Calculus Tutors in Washington DC, Biology Tutors in San Diego

Popular Courses & Classes

GRE Courses & Classes in Washington DC, GRE Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in Chicago, SSAT Courses & Classes in Seattle, Spanish Courses & Classes in Denver, ACT Courses & Classes in Chicago, MCAT Courses & Classes in Atlanta, ACT Courses & Classes in Philadelphia, ISEE Courses & Classes in Dallas Fort Worth, SSAT Courses & Classes in Washington DC

Popular Test Prep

ACT Test Prep in Philadelphia, GRE Test Prep in Dallas Fort Worth, MCAT Test Prep in Philadelphia, SAT Test Prep in New York City, SSAT Test Prep in Phoenix, ISEE Test Prep in Houston, GMAT Test Prep in Phoenix, MCAT Test Prep in San Francisco-Bay Area, ACT Test Prep in Boston, GMAT Test Prep in Seattle

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Topology : Introduction

Study concepts, example questions & explanations for Topology

All Topology Resources

Example Questions

Example Question #1 : Betti Numbers

Example Question #2 : Betti Numbers

Example Question #3 : Betti Numbers

Example Question #2 : Betti Numbers

Example Question #3 : Betti Numbers

Example Question #1 : Euler Characteristic

Example Question #2 : Euler Characteristic

Example Question #3 : Euler Characteristic

Example Question #3 : Introduction

Example Question #4 : Euler Characteristic

All Topology Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: