# Topology ### What does it mean to put a topology on $\mathbb{R}^n$? * See this video [here](https://www.youtube.com/watch?v=jkh5dcddNCU) * Note: A great concrete example of placing a topology on a space is found in pattern find. We start off with a space of patterns (points in a set). We then impose a structure (a topology) via the relationship of parent and child. We then have an idea of how close points are in a space. * [Wikipedia article](https://en.wikipedia.org/wiki/Topological_space) * Intuition for topology * Zoom in arbitrarily deep (to infinitesimal scale) until you arrive at a local patch of euclidean space * The standard topology on $\mathbb{R}^n$ is just a topological structure that we are going to put on these abstract sets so that overall they mirror the topological structure of real spaces. * In order to define a standard topology on $\mathbb{R}^n$ we will need to make use of two things: 1. Make use of the fact that the real numbers have a **field algebraic** structure on it, meaning we can add and multiply any two real numbers. 2. Consider that the real numbers has an **order** on it. * We must ### Consider $\mathbb{R}^2$ If we start with $\mathbb{R}^2$: $\mathbb{R}^2 = \{ (x_1, x_2) \; \mid \; x_i \in \mathbb{R}\}$ We simply have a set of tuples that were created via the [cartesian-product](cartesian-product.md) of $\mathbb{R}$: $\mathbb{R}^2 = \mathbb{R} \times \mathbb{R} $ This *set* has no structure imposed on it yet; in other words, it has no definition of closeness between the points. Our goal is to impose a structure on this set that mimics that of the real world. --- References * [The Standard Topology on $\mathbb{R}^n$ Part 1](https://www.youtube.com/watch?v=jkh5dcddNCU)