Problem 19 - Algebraic Topology

Question

Show that the subspace of \(\mathbb{R}^{3}\) that is the union of the spheres of radius \(1 / n\) and center \((1 / n, 0,0)\) for \(n=1,2, \cdots\) is simply- connected.

Step-by-Step Solution

Answer

The union of these decreasing spheres in \(\mathbb{R}^3\) is simply-connected because it is path-connected and each loop can be contracted to a point.

Step 1: Understanding the Problem

We need to show that the given subspace in \(\mathbb{R}^3\) is simply-connected. A space is simply-connected if it is path-connected and every loop in the space can be continuously contracted to a point.

Step 2: Identify Path-Connectedness

The subspace consists of spheres of decreasing radius with centers on the x-axis. Each sphere is path-connected because, for any two points on the same sphere, there is a continuous path connecting them within the sphere. Therefore, each individual sphere is path-connected.

Step 3: Union of Path-Connected Spaces

To show the union of these spheres is path-connected, observe that for any two points in different spheres (say on the \(n\)-th and \(m\)-th spheres), a path can be constructed by going from the point on the \(n\)-th sphere to its center, along the x-axis to the center of the \(m\)-th sphere, and finally to the point on the \(m\)-th sphere. Thus, the entire subspace is path-connected.

Step 4: Loop Contractibility

Consider a loop in the subspace, which can be contained in a finite number of spheres due to the decreasing radii as \(n\) increases. In \(\mathbb{R}^3\), each spherical component is simply-connected, meaning any loop on a sphere can be contracted to a point within that sphere. Therefore, any loop in the union subspace can also be contracted to a point by crossing between a finite number of simple-connected spheres.

Step 5: Conclusion on Simple-Connectedness

Since the union of these spheres is both path-connected and any loop can be shrunk to a point due to \(\mathbb{R}^3\)'s properties and the construction, the subspace is simply-connected.

Key Concepts

Path-Connectedness

Path-Connectedness is a foundational concept in topology that focuses on how any two points in a given space can be joined by a continuous path. In simpler terms, imagine you have two distinct locations on a map, path-connectedness means that you can draw a route from one point to the other without lifting your pencil.

In the context of the given exercise, each sphere with a fixed radius is a path-connected space. This means for any two points on a single sphere, a continuous path exists within that sphere that connects these points.

• Each sphere has a consistent center and radius, providing an easy way to navigate from one point to another.
• The symmetry of a sphere guarantees path-connectedness within itself.

To achieve path-connectedness in the union of all these spheres, we have to consider more than individual spheres. Since each sphere is centered on the x-axis, connecting points on different spheres involves moving along this axis. This idea is crucial for the entire subspace to be path-connected.

Loop Contractibility

Loop Contractibility involves making loops in a space disappear by gradually shrinking them to a single point. It is an essential property for a space to be considered simply-connected.

In simpler terms, imagine taking a loop or circle in the air and slowly pinching it until it becomes a single dot. In the exercise, this is applied to loops on and across the spheres within the subspace.

• Each individual sphere in our subspace is simply-connected in \(\mathbb{R}^3\), meaning a loop on a sphere can contract to its center point.
• The whole subspace, made from these connecting spheres, allows any loop to contract by utilizing the property that individual spheres themselves allow loop contraction.

When a loop encompasses several spheres, the contraction can cross sphere limits, since each sphere is a safe space for contracting loops. Hence, overall, any loop in the entire subspace can be sufficed to a single point.

Spheres in Topology

Spheres are fundamental shapes in topology with fascinating properties, particularly in their multi-dimensional aspects. They are not only visually captivating but also hold significant theoretical value.

Each sphere in the exercise is centered at \((1/n, 0, 0)\) on the x-axis, and the union of these spheres creates a larger intricate structure in \(\mathbb{R}^3\).

• Each of these spheres has a unique radius \(1/n\), ensuring they decrease in size as \(n\) increases.
• These spheres overlap strategically, as their centers are along the x-axis, lending themselves to path-connectedness by moving from one sphere to another along this axis.

Topology examines these properties and encourages seeing beyond just surface-level geometry. Here, spheres aid in understanding more complex structures by providing simple-connected components that contribute to the nature of the entire subspace.