Date: September 17 2022
Summary: A brief understanding of my knowledge on topological spaces
Keywords: ##summary #algebraic #topology #space #topological #archive
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.
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, , on a nonempty set is a collection of subsets of that also belong to . These subsets are referred to as "open sets" where:
and $X$ are open. Also can be defined as $\emptyset, X \in \tau$
Union of any number of open sets within also belong to
Intersection of a defined number of open sets in also belong to
If these conditions are met, then a topology on is often written as . If or the fact that we are working with topologies is obvious, then one can refer to as the "topology on ".
NOTE: If there are two sets, and , the notation (read as "A drop B") is equal to . This notation is shorthand for . In this case, this set would be known as a closed set under .
If then the trivial topology is .
NOTE: In this case, is referred to as a "family". A "family" is a set of subsets for a given set (in this case, ).
If then the family forms another topology.
NOTE: "General" in the header for this section just refers to possible subsets that can be formed out of the set to form another topology. Additional subsets could be , etc.
If , the discrete topology of is the power set of which is the family . consists of all possible subsets of .
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], [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]
Zelko, Jacob. What Is a Topological Space?. https://jacobzelko.com/09172022051900-what-topology. September 17 2022.