the cedar ledge

What Is a Topological Space?

Date: September 17 2022

Summary: A brief understanding of my knowledge on topological spaces

Keywords: ##summary #algebraic #topology #space #topological #archive

Bibliography

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Table of Contents

    1. Reading Motivation
    2. Topological Spaces
    3. Examples of Topologies on a Set XX
      1. Trivial (Indiscrete) Topology on XX
      2. General Topology on XX
      3. The Discrete Topology of XX
  1. How To Cite
  2. References:
  3. Discussion:

Reading Motivation

I wanted to learn more about algebraic topology so as to further understand the notion of cross products. Cross products are albeit a simple mechanism but I was curious to see the notion of them pop up so much. Much more so seeing the idea propagate to topologies which sparked my interested about topologies.

Topological Spaces

A topological space, to my understanding is a kind of embellished set. Embellished in the sense that there exists an underlying set wherein a topology enriches the set. In short, a topology, τ\tau, on a nonempty set XX is a collection of subsets of XX that also belong to τ\tau. These subsets are referred to as "open sets" where:

\emptyset

and $X$ are open. Also can be defined as $\emptyset, X \in \tau$

  1. Union of any number of open sets within τ\tau also belong to τ\tau

  2. Intersection of a defined number of open sets in τ\tau also belong to τ\tau

If these conditions are met, then a topology τ\tau on XX is often written as (X,τ)(X, \tau). If τ\tau or the fact that we are working with topologies is obvious, then one can refer to (X,τ)(X, \tau) as the "topology on XX".

NOTE: If there are two sets, A={1,2,3}A = \{1, 2, 3\} and B={2,3,4}B = \{2, 3, 4\}, the notation A\BA \backslash B (read as "A drop B") is equal to A\B={1}A \backslash B = \{1\}. This notation is shorthand for ABcA \cap B^{c}. In this case, this set would be known as a closed set under (X,τ)(X, \tau).

Examples of Topologies on a Set XX

Trivial (Indiscrete) Topology on XX

If X={1,2,3,4}X = \{1, 2, 3, 4\} then the trivial topology is τ={{},{1,2,3,4}}={,X}}\tau = \{\{\}, \{1, 2, 3, 4\}\} = \{\emptyset, X\}\}.

NOTE: In this case, τ={,X}}\tau = \{\emptyset, X\}\} is referred to as a "family". A "family" is a set of subsets for a given set (in this case, XX).

General Topology on XX

If X={1,2,3,4}X = \{1, 2, 3, 4\} then the family τ={,{2},{1,2},{2,3},{1,2,3},X}\tau = \{\emptyset, \{2\}, \{1, 2\}, \{2, 3\}, \{1, 2, 3\}, X\} forms another topology.

NOTE: "General" in the header for this section just refers to possible subsets that can be formed out of the set XX to form another topology. Additional subsets could be {1},{1,4}\{1\}, \{1, 4\}, etc.

The Discrete Topology of XX

If x={1,2,3,4}x = \{1, 2, 3, 4\}, the discrete topology of XX is the power set of XX which is the family τ=(X)\tau = \wp(X). (X)\wp(X) consists of all possible subsets of XX.

A simple way of calculating the power set of this topology is with the Julia snippet:

using Combinatorics

℘ = powerset
X = [1, 2, 3, 4]
℘(X) |> collect

Which would give the following possible subsets of X:

[],
[1],
[2],
[3],
[4],
[1, 2],
[1, 3],
[1, 4],
[2, 3],
[2, 4],
[3, 4],
[1, 2, 3],
[1, 2, 4],
[1, 3, 4],
[2, 3, 4],
[1, 2, 3, 4]

How To Cite

Zelko, Jacob. What Is a Topological Space?. https://jacobzelko.com/09172022051900-what-topology. September 17 2022.

References:

Discussion:

CC BY-SA 4.0 Jacob Zelko. Last modified: January 17, 2023. Website built with Franklin.jl and the Julia programming language.