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MTH402Sciences3 Unitsintermediate

General Topology II

This course presents the concepts of topology, which include separability, compactness and connectedness. Various results were proved with sufficient examples to guide learners. It covers topological spaces, separability axioms, category, separability, compactness, connectedness, homotopy relations, and simple connected spaces. The course aims to equip students with the ability to apply topological concepts to other fields of Mathematics.

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65h
Study Time
13
Weeks
5h
Per Week
advanced
Math Level
Course Keywords
TopologySeparabilityCompactnessConnectednessHomotopy

Course Overview

Everything you need to know about this course

Course Difficulty

Intermediate Level
Builds on foundational knowledge
65%
intermediate
∑
Math Level
Advanced Math
📖
Learning Type
Theoretical Focus

Course Topics

Key areas covered in this course

1

Topological Spaces

2

Separation Axioms

3

Category

4

Separability

5

Compactness

6

Connectedness

7

Homotopy Relations

8

Simple Connected Spaces

Total Topics8 topics

Requirements

Knowledge and skills recommended for success

Real Analysis

Set Theory

Abstract Algebra

💡 Don't have all requirements? Don't worry! Many students successfully complete this course with basic preparation and dedication.

Assessment Methods

How your progress will be evaluated (2 methods)

Tutor Marked Assignment

Comprehensive evaluation of course material understanding

Written Assessment

Final Examination

Comprehensive evaluation of course material understanding

Written Assessment

Career Opportunities

Explore the career paths this course opens up for you

Data Analyst

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Research Mathematician

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Theoretical Physicist

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Software Developer

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Financial Analyst

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Industry Applications

Real-world sectors where you can apply your knowledge

Data ScienceFinancial ModelingNetwork AnalysisImage ProcessingTheoretical Physics

Study Schedule Beta

A structured 13-week journey through the course content

Week
1

Module 1: Topological Spaces

3h

Unit 1: Concepts of Topological Spaces

3 study hours
  • •Read the course guide.
  • •Familiarize yourself with the course objectives and competencies.
  • •Understand the structure of the modules and units.
Week
2

Module 1: Topological Spaces

4h

Unit 1: Concepts of Topological Spaces

4 study hours
  • •Study the definitions of topological spaces.
  • •Work through examples of basic concepts.
  • •Understand the basis for topology and subspace topology.
Week
3

Module 1: Topological Spaces

4h

Unit 1: Concepts of Topological Spaces

4 study hours
  • •Continue studying concepts of topological spaces.
  • •Solve self-assessment exercises.
  • •Review the summary and conclusion.
Week
4

Module 1: Topological Spaces

4h

Unit 2: Separation Axioms

4 study hours
  • •Define Hausdorff space and state its properties.
  • •Understand the separation axioms.
  • •Prove that in a Hausdorff space, every point set is closed.
Week
5

Module 1: Topological Spaces

4h

Unit 2: Separation Axioms

4 study hours
  • •Define a convergent sequence and show that in a Hausdorff space, the limit is unique.
  • •Prove that every metric topology is Hausdorff.
  • •Know five separation axioms and their properties.
Week
6

Module 1: Topological Spaces

4h

Unit 2: Separation Axioms

4 study hours
  • •Complete self-assessment exercises.
  • •Review the summary and conclusion.
  • •Read the references for further understanding.
Week
7

Module 2: Separability, Compactness and Connectedness

5h

Unit 1: Category and Separability

5 study hours
  • •Identify dense sets and nowhere dense sets.
  • •Identify sets of first and second categories.
  • •Define separable spaces.
Week
8

Module 2: Separability, Compactness and Connectedness

5h

Unit 1: Category and Separability

5 study hours
  • •State the first and second countability axioms.
  • •Identify first and second countable spaces.
  • •State and prove the sequence lemma and its converse.
Week
9

Module 2: Separability, Compactness and Connectedness

5h

Unit 2: Compact Sets and Spaces

5 study hours
  • •Give the definition of Covers and subcovers.
  • •Define compact sets, subsets and compact spaces.
  • •Give the sequential characterization of compactness.
Week
10

Module 2: Separability, Compactness and Connectedness

5h

Unit 2: Compact Sets and Spaces

5 study hours
  • •Identify sequentially, countably and locally compact sets.
  • •Solve self-assessment exercises.
  • •Review the summary and conclusion.
Week
11

Module 2: Separability, Compactness and Connectedness

5h

Unit 3: Connectedness

5 study hours
  • •Differentiate between connected sets and separated spaces.
  • •Define connected spaces.
  • •Understand the connectedness to the real line.
Week
12

Module 2: Separability, Compactness and Connectedness

5h

Unit 3: Connectedness

5 study hours
  • •Identify the connected components of a given space.
  • •Identify locally connected spaces.
  • •Know and use of the concept of path connectedness.
Week
13

Module 3: Homotopy Relations

12h

Unit 1: Homotopy of Paths

6 study hours
  • •Review all modules and units.
  • •Work on assignments and tutor-marked assignments.
  • •Prepare for the final examination.

Unit 2: Simple Connected Spaces

6 study hours
  • •Understand the concepts of homotopic paths.
  • •Distinguish between paths and loops.
  • •Understand when a topological space is simply connected.

This study schedule is in beta and may not be accurate. Please use it as a guide and consult the course outline for the most accurate information.

Course PDF Material

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Study Tips & Exam Preparation

Expert tips to help you succeed in this course

1

Thoroughly review all definitions and theorems from each unit.

2

Practice solving problems from the self-assessment exercises in each unit.

3

Create concept maps linking topological spaces, separation axioms, and connectedness.

4

Focus on understanding the proofs of key theorems, such as the Heine-Borel theorem and the intermediate value theorem.

5

Review all tutor-marked assignments and address any areas of weakness.

6

Allocate sufficient time to practice applying the concepts to concrete examples.

7

Form study groups to discuss challenging concepts and problem-solving strategies.

8

Prioritize understanding the relationships between different topological properties, such as compactness, connectedness, and separability.

9

Practice constructing counterexamples to disprove false statements and deepen your understanding of the material.

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