Notes on Topology

Working note — Part of my personal research notebook. Entries may be incomplete and are revised over time.

Notes

pre-requisite: real-analysis

(1) Some random notes:

• real analyis is a sufficient background to get started, and topology is a natural next step
• without at least an intermediate knowledge in set theory, it can be hard to delve into more interesting stuff in point-set Topology
• “irreducible core” of the subject: topological spaces, connectedness, compactness, and the countability and separation axioms
• what is the point of generalization?
• Course outline Part 1:
  • set theory and logic,
  • topological spaces and continouos function,
  • connectedness and compactness,
  • countability
  • separation axioms

(2) Topology without tears

• topological notions like compactness, connectedness and denseess are as basic to mathematician of today as sets and functions were to those of last century.
• branches of topology:
  • general topology (point-set topology)
  • algebraic topology
  • differential topology
  • topological algebra
• Point-set topology is the door to the study of others
• in mathematics “or” is not exclusive, a or b (a and b being true at the same time)

• only by working through a good number of exercises will you master this course

  Chapter 1 - topological spaces

  • we begin with the rules of topology
  • definitions
    • finite topological spaces
    • discrete spaces: given a non-empty set X, if T is a collection of all the subsets of X, then T is said to be the discrete topology. (X,T) is called the topological space
      • there is an infinite number of discrete spaces one for each set X
    • indiscrete spaces: given a non-empty set X, if T={X,0} (0 is null set) then T is called the indiscrete topology and (X,T) is said to be an indiscrete space
    • spaces with the finite-closed topology
  • topology is an axiomatic subject: each theorem or proposition represents a new level of knowledge and must be firmly anchored to the previous level, we attache the new level to the previous one using a proof
  • what is a mathematical proof: a sequence of arguments, begins with information you are given, proceeds by logical argument and ends with what you are asked to prove
  • begin a proof by writing down the information you are given
  • state what you asked to prove
  • if technical terms are given, define them