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MATH-510: Topology

A continuous deformation (a type of homeomorphism) of a mug into a doughnut (torus) and back.

Topology is an important branch of mathematics that studies all the “qualitative” or “discrete” properties of continuous objects such as manifolds, i.e. all the properties that aren't changed by any continuous transformations except for the singular (infinitely extreme) ones.

In this sense, topology is a vital arbiter in the “discrete vs continuous” wars. The very existence of topology as a discipline shows that “discrete properties” always exist even if you only work with continuous objects. On the other hand, topology always assumes that these features are “derived” – they're some of the properties of objects and these objects are deeper and that may have many other, continuous properties, too. The topological, discrete properties of these objects are just projections or caricatures of the “whole truth”. (continue reading ...)

This is an introductory course in topology, giving the basics of the theory.

Topological spaces, bases and sub-bases, first and second axiom of countability, separability, continuous functions and homeomorphism, finite product space. Separation axioms $(T_0, T_1, T_2)$. Regular spaces, completely regular spaces, normal spaces, compact spaces, connected spaces.

  1. Sheldon Davis, Topology, McGraw-Hill Science/Engineering/Math, 2004.
  2. Seymour Lipschutz, Schaums Outline of General Topology, McGraw-Hill, 2011.
  3. James Munkres, Topology (2nd Edition), Prentice Hall, 2000.
  4. G.F. Simmons, Introduction to Topology and Modern Analysis, Tata McGraw-Hill, 2004. (link)
  5. Stephen Willard, General Topology, Dover Publications, 2004. (link)
  6. M.A. Armstrong, Basic Topology, Springer, 2010.

Brain teasers: dare to think!

Some questions are given below. These should be considered as sample and thousands of such questions can be created or constructed but if you understand the basic knowledge and definitions then there is no problem to answer such type of questions.

  1. Is it possible to construct a topology on every set?
  2. Give an example of open set in $\mathbb{R}$ with usual topology, which is not an open interval.
  3. Let $X=\{a\}$. Then what are the differences between discrete topology, indiscreet topology and confinite topology on $X$?
  4. Let $X$ be a non-empty finite set. Then what is the difference between discrete and cofinite toplogy on $X$.
  5. Let $\tau$ be a cofinite toplogy on $\mathbb{N}$. Then write any three element of $\tau$.
  6. Let $(\mathbb{Z}, \tau)$ be a cofinite topological spaces.
    • Is $\mathbb{N}$ open in $\tau$?
    • Is $A=\{\pm 100,\pm 101, \pm 102, \ldots \}$ open in $\tau$?
    • Is $E=\{0,\pm 2,\pm 4,\ldots\}$ open in $\tau$?
    • Is set of prime open in $\tau$?
    • Is $B=\{1,2,3,\ldots,99\}$ closed in $\tau$?
    • Is $C=\{10^{10}+n : n \in \mathbb{Z} \}$ open in $\tau$?
  7. Write the closure of the set $S=\left\{1+\frac{1}{n}: n \in \mathbb{N} \right\}$ in usual topology on $\mathbb{R}$.
  8. What is the closure of the set $T=\{1,2,3,4,5 \}\cup (6,7) \cup (7,8] $ in usual topology on $\mathbb{R}$?
  9. What is the closure of the set $U=\{101,102,103,\ldots,200\}$ in a cofinite toplogy constructed on $\mathbb{Q}$?

Presentation

Introduction to Topology Download Presentation (1.6MB)

Selected questions

Selected questions from chapter 05 of [2], that is, Schaums Outline of General Topology.

Starting from page 73. (total 43 questions)

01, 02, 03, 04, 05, 10, 11, 12, 13, 14, 15, 
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 34, 36, 37, 40, 43, 44,
46, 47, 53, 61, 63, 68, 75, 78, 85, 86. 


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