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MTH401Sciences3 Unitsintermediate

General Topology I

This course introduces students to the fundamental concepts of general topology. It covers metric spaces, topological notions, geometric properties, sequences, continuity, completeness, compactness, and connectedness. Students will learn to define and apply these concepts, verify examples, and prove basic theorems. The course provides a foundation for further study in real analysis, complex analysis, and functional analysis.

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156h
Study Time
13
Weeks
12h
Per Week
intermediate
Math Level
Course Keywords
TopologyMetric SpacesCompactnessConnectednessContinuity

Course Overview

Everything you need to know about this course

Course Difficulty

Intermediate Level
Builds on foundational knowledge
65%
intermediate
📊
Math Level
Moderate Math
📖
Learning Type
Theoretical Focus

Course Topics

Key areas covered in this course

1

Metric Spaces

2

Topological Notions

3

Open and Closed Sets

4

Sequences and Convergence

5

Continuity

6

Completeness

7

Compactness

8

Connectedness

Total Topics8 topics

Requirements

Knowledge and skills recommended for success

Real Analysis

Calculus

💡 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 (3 methods)

assignments

Comprehensive evaluation of course material understanding

Written Assessment

tutor-marked assessments

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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Statistician

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Financial Analyst

Apply your skills in this growing field

Software Engineer

Apply your skills in this growing field

Industry Applications

Real-world sectors where you can apply your knowledge

Data ScienceFinancial ModelingImage ProcessingNetwork AnalysisOptimization

Study Schedule Beta

A structured 13-week journey through the course content

Week
1

Module 1: Metric Spaces

5h

Unit 1: Metric Spaces

5 study hours
  • •Define metric space and its properties.
  • •Verify if a given function is a metric.
  • •Identify Euclidean metric on R^n.
  • •Solve problems related to metric spaces.
Week
2

Module 2: Topological Notions; Geometric properties

5h

Unit 2: Topological Notions; Geometric properties

5 study hours
  • •Define open balls, closed balls, and spheres in metric spaces.
  • •Compute open and closed balls for given metric spaces.
  • •Identify open and closed sets.
  • •Determine interior and limit points of sets.
Week
3

Module 3: Sequences

5h

Unit 3: Sequences

5 study hours
  • •Define convergent sequence and give examples.
  • •Show that a sequence is convergent or not.
  • •Define subsequence of a sequences.
  • •Define a Cauchy sequence.
Week
4

Module 4: Continuity

5h

Unit 4: Continuity

5 study hours
  • •Define continuity at a point.
  • •Show that a function is continuous at a given point.
  • •Give a sequential characterization of a continuity.
  • •Prove some basic theorem on continuity.
Week
5

Module 5: Completeness

5h

Unit 5: Completeness

5 study hours
  • •Define a complete metric space and give examples.
  • •Prove theorems concerning complete metric spaces.
  • •Understand Banach Contraction Mapping Principle.
Week
6

Module 6: Compactness

5h

Unit 6: Compactness

5 study hours
  • •Understand the definition of compactness
  • •Give some examples of compactness.
  • •State and prove some important theorem on compactness
  • •State the characteristics of a continuous function defined on a compactness.
Week
7

Module 7: Connectedness

5h

Unit 7: Connectedness

5 study hours
  • •Define and explain the of connectedness in a metric space.
  • •Give examples and state some basic properties of connected sets.
  • •See a special property of a continuous function defined on a connected space.
  • •Prove the intermediate value theorem.
Week
8

Module 1: Metric Spaces

8h

Unit 1: Metric Spaces

4 study hours
  • •Review Metric Spaces
  • •Solve TMAs questions

Unit 2: Topological Notions; Geometric properties

4 study hours
  • •Review Topological Notions
  • •Solve TMAs questions
Week
9

Module 3: Sequences

8h

Unit 3: Sequences

4 study hours
  • •Review Sequences
  • •Solve TMAs questions

Unit 4: Continuity

4 study hours
  • •Review Continuity
  • •Solve TMAs questions
Week
10

Module 5: Completeness

8h

Unit 5: Completeness

4 study hours
  • •Review Completeness
  • •Solve TMAs questions

Unit 6: Compactness

4 study hours
  • •Review Compactness
  • •Solve TMAs questions
Week
11

Module 7: Connectedness

8h

Unit 7: Connectedness

8 study hours
  • •Review Connectedness
  • •Solve TMAs questions
Week
12

Final Revision

8h

Final Revision

8 study hours
  • •Complete all TMAs
  • •Prepare for examination
Week
13

Final Revision

8h

Final Revision

8 study hours
  • •Complete all TMAs
  • •Prepare for examination

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

Focus on definitions and examples of metric spaces, topological properties, and convergence.

2

Practice proving basic theorems related to continuity, completeness, and compactness.

3

Review and understand the statements and applications of the Banach Contraction Mapping Principle.

4

Work through all examples and exercises in the course material, paying close attention to TMAs.

5

Create concept maps linking metric spaces, topological notions, sequences, and continuity.

6

Allocate sufficient time to review and consolidate each module before moving on to the next.

7

Prioritize understanding the core definitions and theorems over memorizing proofs.

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