MTH634 ASSIGNMENT NO. 2 FALL 2022 || 100% RIGHT SOLUTION || TOPOLOGY || BY VuTech

MTH634 ASSIGNMENT NO. 2 FALL 2022

 KINDLY, DON’T COPY PASTE

MTH634 ASSIGNMENT NO. 2 FALL 2022 || 100% RIGHT SOLUTION || TOPOLOGY || BY VuTech

Visit Website For More Solutions

 www.vutechofficial.blogspot.com

SEND WHATSAPP OR E-MAIL FOR ANY QUERY

0325-6644800

kamranhameedvu@gmail.com

Fall 2022            MTH634: Topology

Assignment No. 2 (Lectures No. 12 to 17-Topic# 36 to 61)     

Maximum Marks: 10

 Due Date: Thursday, January 19, 2022

 

Please read the following instructions carefully before attempting the solution of this assignment:

(1)   To solve this assignment, you should have good command over Topic# 36 to 61 (Lecture No. 12 to 17).

(2)   Try to consolidate your concepts that you learn in the lectures with these questions.

(3)   Upload your assignment properly through VULMS. No Assignment will be accepted through emails.

(4)   First of all, you should  the uploaded question file. Next, solve the given questions (i.e., type the solution) on it. Finally, submit your own solution file in VULMS.

Ø  Understand the problem then solve it according to the statements (requirements) of the questions.

Ø  If you do not solve the questions by the method that is asked (required) in the problems then the marks will be deducted (or could be zero).

(5)   Do not use colorful backgrounds in your solution files.

Ø  All the students are directed to use the font and style of text as is used in this document i.e. the font size/style should be preferably 12 Times New Roman with black font color.

Ø  The assignments should be zoomed in at 100%.

(6)   Use MathType or Equation Editor etc. for writing the proper mathematical symbols/expressions and equations.

Ø  In MS-Word, to type an equation from scratch, press Alt += on your keyboard or choose Insert --> Equation and select Insert New Equation from the bottom of the built in equation gallery. This inserts an equation placeholder where you can type your equation. It is recommended to visit the following link for more detail:

Ø  https://support.microsoft.com/en-us/office/write-an-equation-or-formula-1d01cabc-ceb1-458d-bc70-7f9737722702#ID0EAACAAA=Write_new_equation

Ø  Remember that you are supposed to submit your assignment only in the MS-Word format, any other format like scanned, images, PDF, MS-Excel, HTML etc. will not be accepted.

Ø  Corrupt files will be given zero marks.

(7)   This is an individual assignment (not a group assignment). So keep in mind that you are supposed to submit your own, self-made and different assignments even if you discuss the questions with your class fellows. All the similar assignments (even with some meaningless modifications) will be awarded zero marks and no excuse will be accepted. This is your responsibility to keep your assignment safe from others.

Up to 50% marks might be deducted for those assignments which are received after the due date.

Question:                                                                                                              Marks: 10

 

Q#1:

 Let X ={a,b,c,d,e}   and let A={{a,b,d}, {d,c}, {c,e}, X} ba a subbase. Find the base B generated by A. Also, find the topology generated by B. 

`\alpha` = {{a,b,c},{c,d},{d,e}}   be a subbase. Find the base  `\beta` generated by  `\alpha`. Also, find the topology generated by   `\beta`.

Solution:

 

Part-1:

Let X = {a, b, c, d, e} and let A = {{a, b, d}, {d, c}, {c, e}, X} be a subbase. To find the base B generated by A, we need to take all possible unions of elements in A.

 

B = {A ∪ B | A, B ∈ A}

 

= {({a, b, d} ∪ {d, c}), ({d, c} ∪ {c, e}), ({c, e} ∪ X), ({a, b, d} ∪ {d, c} ∪ {c, e} ∪ X)}

 

= {{a, b, c, d}, {c, d, e}, {a, b, c, d, e}}

 

The topology generated by B is the collection of all subsets of X that can be formed by taking arbitrary unions of elements in B. In other words, it is the collection of all subsets of X that can be formed by taking arbitrary unions of the sets in B.

Topology = {S ⊆ X | S = U₁ ∪ U₂ ∪ ... ∪ Uₙ, U₁, U₂, ..., Uₙ ∈ B}

Part-2: 

            Let `\alpha` = {{a,b,c},{c,d},{d,e}} be a subbase. To find the base `\beta` generated by `\alpha`, we need to take all possible unions of elements in `\alpha`.

`\beta` = {`\alpha` ∪ `\beta` | `\alpha`, `\beta` ∈ `\alpha`}

 

= {({a,b,c} ∪ {c,d}), ({c,d} ∪ {d,e}), ({a,b,c} ∪ {c,d} ∪ {d,e})}

 

= {{a,b,c,d}, {c,d,e}, {a,b,c,d,e}}

 

The topology generated by `\beta` is the collection of all subsets of X that can be formed by taking arbitrary unions of elements in `\beta`. In other words, it is the collection of all subsets of X that can be formed by taking arbitrary unions of the sets in `\beta`.

 

Topology = {S ⊆ X | S = U₁ ∪ U₂ ∪ ... ∪ Uₙ, U₁, U₂, ..., Uₙ ∈ `\beta`}

 

 Visit Website For More Solutions

 www.vutechofficial.blogspot.com

KINDLY, DON’T COPY PASTE

SUBSCRIBE, SHARE, LIKE AND COMMENTS FOR MORE UPDATES

SEND WHATSAPP OR E-MAIL FOR ANY QUERY

0325-6644800

kamranhameedvu@gmail.com