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MathidiaMathematics Academy
Topology
Lesson 11 of 75
Introduction

Topological spaces

Objective. Work from open-set axioms and generated topologies.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A topological space is a set X together with a collection ฯ„ of subsets called open sets, satisfying that โˆ… and X are open, arbitrary unions of open sets are open, and finite intersections of open sets are open. This abstracts the idea of 'nearness' without needing distance: any subcollection of ฯ„ can serve as a basis, and a topology generated by a collection ๐’ฎ is the smallest topology containing ๐’ฎ, built from finite intersections (forming a basis) then arbitrary unions. For example, the discrete topology on X takes ฯ„ to be all subsets of X, while the indiscrete topology has only ฯ„={โˆ…,X}.

Worked example

On X={a,b,c}, which collection is a valid topological space's topology: {โˆ…,X,{a}} or {โˆ…,X,{a},{b}}?

Options
  • {โˆ…,X,{a}}
  • {โˆ…,X,{a},{b}}
  • Both are valid topologies
  • Neither is a valid topology
  1. 1Check {โˆ…,X,{a}}: unions and intersections of these three sets stay within the collection
  2. 2This satisfies all three topology axioms
  3. 3Check {โˆ…,X,{a},{b}}: union {a}โˆช{b}={a,b} is missing from the collection
  4. 4So only {โˆ…,X,{a}} is a valid topology

Answer: {โˆ…,X,{a}}

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Practice

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Common mistake. Students think infinite intersections of open sets must be open, but in โ„ with the usual topology, โˆฉ_{n=1}^{โˆž}(โˆ’1/n,1/n) = {0}, which is not open, showing the axioms only guarantee finite intersections stay open.

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