Question

The one-point compactification of \(\mathbb{R}^{n}\) is homeomorphic to the \(n\)-sphere \(\\{x \in\) \(\left.\mathbb{R}^{n+1}:|x|=1\right\\}\).

Short answer

The one-point compactification of \(\mathbb{R}^n\) is homeomorphic to the \(n\)-sphere.

Step-by-step solution

Understanding Compactification

The one-point compactification of a space \(X\) refers to adding a "point at infinity" to \(X\), denoted by \(X^* = X \cup \{\infty\}\). For \(\mathbb{R}^n\), which is not compact, the one-point compactification turns it into a compact space.

Topology of Compactification

The topology on \(\mathbb{R}^n \cup \{\infty\}\) is such that open sets in \(\mathbb{R}^n\) remain open, and the sets of the form \(\{x \in \mathbb{R}^n: |x| > R\} \cup \{\infty\}\) for some \(R > 0\) are additional open sets, which include the point at infinity.

Defining the Sphere

The \(n\)-sphere, \(S^n = \{ x \in \mathbb{R}^{n+1} : |x| = 1 \}\), is a compact, closed subset of \(\mathbb{R}^{n+1}\). It is the boundary of the unit ball in \(\mathbb{R}^{n+1}\).

Constructing the Homeomorphism

To show \(\mathbb{R}^n \cup \{\infty\}\) is homeomorphic to \(S^n\), consider the stereographic projection from \(S^n\) minus the north pole \((0,\ldots,0,1)\) to \(\mathbb{R}^n\). The map from \(S^n\) to \(\mathbb{R}^n \cup \{\infty\}\) is defined by this projection, with the newly added point corresponding to the north pole of the sphere.

Verifying Properties of Homeomorphism

This map is continuous, bijective, and its inverse is also continuous. Continuity is assured by the properties of the stereographic projection, compactness of \(S^n\), and openness of \(\mathbb{R}^n\), ensuring it is a homeomorphism.

Key concepts

Compactification

When dealing with mathematical spaces, compactification is like giving a non-compact space a snug enclosure. Imagine you have a vast landscape like \(\mathbb{R}^n\), which stretches endlessly. To make it manageable, we add a single new "point at infinity," rendering the landscape compact. This process is called the one-point compactification, and the resulting space is denoted \(X^* = X \cup \{\infty\}\). This additional point ensures that every sequence (or path) that would otherwise wander off into infinity can be said to converge to this new point. Think of it like finishing a picture by drawing a frame around it, bringing everything neatly together.

Homeomorphism

In topology, homeomorphism is a way to understand when two spaces are essentially the same in shape or structure, even if they might appear different at first glance. It's like saying two objects can be molded into one another without tearing or gluing. For a function to be a homeomorphism, it must be continuous, one-to-one, onto, and have a continuous inverse. Consider the one-point compactification of \(\mathbb{R}^n\) and the \(n\)-sphere: the stereographic projection creates a bridge between them. This map respects all the rules of a homeomorphism, making the seemingly infinite plane and the compact sphere equivalent in topological terms. This means they share the same fundamental properties, despite looking different.

Stereographic Projection

Stereographic projection is a clever method used to project a sphere onto a plane. Visualize projecting points from a sphere, like a globe, onto a flat surface by shining a light from the North Pole through the globe. Every point on the sphere corresponds to a point on the plane, except the North Pole itself, which we can think of as mapping to infinity. This method not only helps in mapping spheres to planes but also elegantly shows a connection between compact spaces, allowing us to understand complex structures better. It's like unfolding a multidimensional map onto a two-dimensional sheet without losing any details about how the points are structured relative to one another.

n-Sphere

An \(n\)-sphere is a generalization of a circle and a sphere to higher dimensions, denoted as \(S^n = \{ x \in \mathbb{R}^{n+1} : |x| = 1 \}\). For example, a 1-sphere is a circle, a 2-sphere is a regular sphere, and so on. The \(n\)-sphere is the boundary of the \((n+1)\)-dimensional unit ball centered at the origin. It's always compact and closed, making it a fundamental object of study in topology. While it can be hard to visualize higher-dimensional spheres, they help extend intuitive geometric ideas to complex scenarios. They provide us with insights into spatial properties and are essential in studying the structure of various mathematical entities.