^^ [[Schutz - 1 - Some Basic Mathematics|1. Some Basic Mathematics]] | [[Schutz - 1.2 - Mappings|1.2 - Mappings]] >> # The Space $\mathbb{R}^n$ and its Topology A point in $\mathbb{R}^n$ is a sequence of $n$ real numbers $(x_1, x_2, \ldots, x_n)$ called an n-tuples of real numbers. We have an intuitive notion of continuity in this space, there are points of $\mathbb{R}^n$ arbitrarily close to any given point, a line joining any two points can be subdivided into arbitrarily many pieces that also join points of $\mathbb{R}^n$. To make the concept of continuity precise in this space, we use its topology, specifically its *local* topology. > **Neighborhood of a point in $\mathbb{R}^n$** > Defining a distance function between two points $\pmb{x} = (x_1, x_2, \ldots, x_n)$ and $\pmb{y} = (y_1, y_2, \ldots, y_n)$: > $ d(\pmb{x}, \pmb{y}) = [(x_1 - y_1)^2 + (x_2 - y_2)^2 + \ldots + (x_n - y_n)^2]$ > The neighborhood of radius $r$ of the points $\pmb{x}$ in $\mathbb{R}^n$ is the set of points $N_r(\pmb{x})$ whose distance from $\pmb{x}$ is less than $r$. See the figure below for an example in $\mathbb{R}^2$, where the interior of the circle is the neighborhood defined by the distance function above. The circle itself is not included. ^37f45e ![[Pasted image 20210122192044.png]] > **Hausdorff property of $\mathbb{R}^n$** > The idea that a line joining any two points of $\mathbb{R}^n$ can be infinitely subdivided can be made more precise by saying that any two points of $\mathbb{R}^n$ have non-intersecting [[Schutz - 1.1 - The Space R^n and its Topology#^37f45e|neighborhoods]]. They will also have neighborhoods that do intersect, but if the neighborhood radius $r$ is small enough, we can make them disjoint. > **Open sets of $\mathbb{R}^n$ and induced topology** > When we use a distance function $d(\pmb{x}, \pmb{y})$ to define neighborhoods and thus open sets in $\mathbb{R}^n$, then we say that $d(\pmb{x}, \pmb{y})$ induces a topology on $\mathbb{R}^n$. > We mean that it has enabled us to define open sets of $\mathbb{R}^n$ which have the following properties: > 1. if $O_1$ and $O_2$ are open, so is their intersection $O_1 \cap O_2$ > 2. the union of any collection of (possibly infinitely many) open sets is also open. > > This requires is to define the empty set $\emptyset$ to be open to obey (1) and $\mathbb{R}^n$ itself as open to obey (2) > In this sense, the distance function allows us to $d(\pmb{x}, \pmb{y})$ make $\mathbb{R}^n$ into a topological space. > Note that the induced topology does **not** depend on the specific form of $d(\pmb{x}, \pmb{y})$. All we need to know is only a notion that the distance between points can be made arbitrarily small and that no two distinct points have zero distance between them ^224639 > **More general definition of neighborhoods** > We can define the neighborhood of a point $\pmb{x}$ to be any set containing an open set containing $\pmb{x}$.