Problem 1 - Analysis Now

Question

Show that the unit circle \(S^{1}\) in \(\mathbb{R}^{2}\) and the unit interval \([0,1]\) both are (Hausdorff) compactifications of \(\mathbb{R} .\) Hint: Use the fact that \(\mathbb{R}\) is homeomorphic to the open interval ] 0,1 [ and (therefore also) homeomorphic to \(\mathrm{S}^{1} \backslash\\{(1,0)\\}\).

Step-by-Step Solution

Answer

Both \([0,1]\) and \(S^1\) compactify \(\mathbb{R}\) by being its dense extensions, with \(\mathbb{R}\) fitting into each via homeomorphisms to \((0,1)\) and \(S^1 \setminus \{(1,0)\}\).

Step 1: Understand Compactifications

A compactification of a space \(X\) is a compact space \(Y\) such that \(X\) is embedded in \(Y\) as a dense subset. A Hausdorff compactification further requires \(Y\) to be Hausdorff. We will show that both \(S^1\) and the unit interval \([0, 1]\) can serve as such extensions for \(\mathbb{R}\).

Step 2: Show \([0,1]\) is a Compactification

\(\mathbb{R}\) can be mapped homeomorphically to \((-\infty, \infty)\), which is homeomorphic to \((0,1)\) via \(x \mapsto \frac{1}{\pi} \arctan(x) + \frac{1}{2}\). The closure in \([0,1]\) is \([0,1]\) itself since \(\mathbb{R}\) is dense in \((0,1)\). Thus, \([0,1]\) serves as the compactification of \(\mathbb{R}\).

Step 3: Prove \(S^1\) is a Compactification

Consider \(S^1 \setminus \{(1,0)\}\), which is homeomorphic to \(\mathbb{R}\). Using the stereographic projection, every point on \(\mathbb{R}\) maps to a unique point on \(S^1\), except \((1,0)\), making \(S^1\) a one-point compactification of \(\mathbb{R}\). Since \(S^1\) is compact and Hausdorff, it is a Hausdorff compactification.

Step 4: Apply the Hint

Since the hint tells us that \(\mathbb{R}\) is homeomorphic to both \((0,1)\) and \(S^1 \setminus \{(1,0)\}\), we've shown the embeddings are dense in \([0, 1]\) and \(S^1\). Through these homeomorphisms, both spaces can thus compactify \(\mathbb{R}\) by including the necessary boundary points \(0\) and \(1\) in \([0,1]\) or the point \((1,0)\) in \(S^1\).

Key Concepts

Hausdorff space

A Hausdorff space is a type of topological space that has a specific separation property. In simple terms, a space is said to be Hausdorff if any two distinct points can be separated by neighborhoods that do not intersect. This means that given two distinct points in such a space, you can find two open sets where each set contains one point and the sets do not overlap.

This property is crucial for ensuring that limits of sequences or nets (generalizations of sequences) are unique. Without the Hausdorff condition, we could have strange situations where sequences could converge to multiple different points, causing ambiguity.

The unit circle, denoted as \(S^1\), is an example of a Hausdorff space. This property is part of what makes \(S^1\) suitable for functions that rely on unique limits, like continuous mappings. Additionally, in analysis and geometry, working with Hausdorff spaces is advantageous because they make certain theorems applicable, such as the uniqueness of limits and continuity.

unit circle

The unit circle \(S^1\) is the set of all points in the plane that are exactly one unit away from a fixed center point, often the origin. It is expressed mathematically as \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}.

In topology, the unit circle is an important structure because it is both compact and a simple example of a closed, bounded set in \mathbb{R}^2\. Being compact means it contains every point it needs to keep sequences from "escaping" to infinity, and this forms part of the essential properties of compactifications when trying to extend real numbers \mathbb{R}\ into a closed set.

The unit circle is also handy as a model of a compact space in studies involving topological functions and transformations due to its simple geometric structure, making it the "perfect circle" for use in theoretical constructs.

homeomorphism

Homeomorphism is a fundamental concept in topology. It refers to a mapping between two spaces that is continuous, bijective, and has a continuous inverse. In essence, two spaces are homeomorphic if they can be reshaped into each other without tearing or gluing.

This concept allows for the generalization of geometric properties while maintaining the spaces' intrinsic topological structure. For instance, \(\mathbb{R}\) and the open interval \( (0, 1) \) can be considered homeomorphic. This means that there exists a continuous function with a continuous inverse that connects these two spaces, preserving their structure.

In the context of the problem, seeing \(\mathbb{R}\) as homeomorphic to certain intervals or subsets allows us to understand how every point in \(\mathbb{R}\) could map uniquely to points in the unit circle \(S^1\) or the interval \[0, 1\]. This property is what makes it possible to use these spaces as compactifications of \(\mathbb{R}\).

dense subset

A dense subset in a topological space is a subset whose closure is the entire space. In simpler terms, every point in the space can be approached arbitrarily closely by points from the dense subset.

This concept is essential for compactifications, as the original space \(\mathbb{R}\) must be densely embedded within the new compact space, whether it's \(S^1\) or \[0, 1\]. The idea is that we want the behavior of the dense subset to represent or "approximate" the entire space.

For example, if we take \(\mathbb{R}\) and view it as densely embedded in \[0, 1\] through a homeomorphic mapping to \((0, 1)\), every point on \[0, 1\] represents a limit point of some sequence from \(\mathbb{R}\). This allows the interval \[0, 1\] to serve as a compactification by adding only the "necessary" limit points, which in this case, are the boundaries 0 and 1. Understanding density helps explain why certain spaces are chosen to compactify more open, infinite spaces like \(\mathbb{R}\).