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

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Table of Contents

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, $\tau$, on a nonempty set $X$ is a collection of subsets of $X$ that also belong to $\tau$. These subsets are referred to as "open sets" where:

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

2. Union of any number of open sets within $\tau$ also belong to $\tau$

3. Intersection of a defined number of open sets in $\tau$ also belong to $\tau$

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

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

Examples of Topologies on a Set $X$

Trivial (Indiscrete) Topology on $X$

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

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

General Topology on $X$

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

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

The Discrete Topology of $X$

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

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.

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