MTH251: Set Topology (Spring 25)

MTH251 Set Topology

Set topology is a branch of mathematics that studies the properties of shapes and spaces that remain unchanged even if they are stretched, twisted, or deformed (without tearing or gluing). It helps us understand concepts like continuity, connectedness, and boundaries.

Imagine a road map 🗺️ where cities are points, and roads connecting them form paths—topology focuses on the connections rather than distances. It is used in Google Maps, networking, physics, and even robotics!

Though it may seem difficult at first, set topology provides a powerful way to study the structure of spaces in mathematics and real life. 😊

What You Will Learn in This Course

✅ Understanding Metric & Topological Spaces

Learn the theory behind metric spaces and topological spaces.

✅ Writing Logical Proofs

Understand how to write step-by-step mathematical proofs using key theorems and properties.

✅ Problem Solving in Topology

Apply topology concepts to solve mathematical problems.

✅ Presenting Solutions Clearly

Write solutions in correct mathematical English with proper logical arguments.

✅ Developing Mathematical Skills

Improve mathematical writing, logical reasoning, and presentation of rigorous proofs.

This course will help you not only understand topology but also grow in your overall mathematical thinking! 😊

Course contents

Preliminaries, Metric spaces: Open and closed sets, convergence, completeness.

Continuous and uniformly continuous mappings. Pseudometrics. Fixed point theorem for metric spaces; Topological Spaces. Open bases and sub-bases. Relative topology, Neighborhood system, Limit points, First and second countable spaces. Separable spaces. Products of spaces, Interior, Exterior, Closure and Frontier in product spaces.

Open and closed maps, Continuity and Homeomorphisms, Quotient spaces; Housdorff spaces, regular, and normal spaces, Urysohn's Lemma; Compact spaces, Tychonoff's theorem and locall compact spaces, Compactness for Metric spaces; Connected spaces, Components of a space, Totally disconnected spaces, Local connectedness, Path-wise connectedness

Topics to cover

Topological spaces: Definitions

  • Define topology on a set.
  • Define open set.
  • What is discrete topological space?
  • What is usual topology on $\mathbb{R}$?
  • What is indiscrete topology?
  • What is cofinite topology or $T_1$-topology?
  • Define closed set.
  • Write three open sets and three closed set of the cofinite topology on $\mathbb{Z}$.
  • Prove that intersection of topological spaces is topological space.
  • Give an example of topological spaces such that their union is not topological space.
  • Define accumulation points or define limit-point.
  • Define derive set.
  • Find the derive set of $A=\{1,2,3,...,20\}$ under the usual topology on $\mathbb{R}$.
  • What is the derive set of $\mathbb{Q}$ under the usual topology on $\mathbb{R}$?
  • Consider the set $A=\left\{1,\frac{1}{2},\frac{1}{3},... \right\}$. Find the derive set of $A$ under usual topology.
  • Define closure of a set.
  • Define dense set.
  • Define interior point, exterior point and boundary point.
  • Under the usual topology on $\mathbb{R}$, find interior and closure of the following sets:
    • (i) $A=\mathbb{N}$ (ii) $B=\{1,2,3,...,100\}$ (iii) $C=[-1,1]$
    • (iii) $D=(0,5]$ (iv) $E=\{1,2,3\}\cup[4,5]$ (vi) $F=(3,10)$ (vii) $G=\mathbb{Q}$
  • Under the discrete topology on $\mathbb{R}$, write interior and closure of the following sets:
    • (i) $A=\{1,2,...,10\}$ (ii) $B=[0,1)$ (iii) $C=\mathbb{Q}$
  • Define relative topology.
  • Let $X=\{a,b,c,d\}$ and $\tau=\{\varphi, X, \{a\}, \{a,c\}\}$. Then find the relative topology of $A=\{c,d\}$.
  • Let $A$ be a subset of topological space $X$. Then prove that $A$ is closed in $X$ iff $A'\subset A$.
  • Let $A$ be a subset of topological space. Then prove that $A\cup A'$ is closed.
  • Let $A$ be a subset of topological space. Then prove that $\overline{A}=A\cup A'$
  • Let $A$ and $B$ be subsets of topological space. Then prove that $\overline{A\cup B}=\overline{A}\cup \overline{B}$.

Assignment

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Resources

Recommended book

  1. Seymour Lipschutz, Schaum's Outline of General Topology, McGraw-Hill, 2011.
  2. James Munkres, Topology (2nd Edition), Prentice Hall, 2000.

Other books

  1. Sheldon Davis, Topology, McGraw-Hill Science/Engineering/Math, 2004.
  2. Seymour Lipschutz, Schaum's Outline of General Topology, McGraw-Hill, 2011.
  3. G.F. Simmons, Introduction to Topology and Modern Analysis, Tata McGraw-Hill, 2004. (link)
  4. Stephen Willard, General Topology, Dover Publications, 2004. (link)
  5. M.A. Armstrong, Basic Topology, Springer, 2010.