Homeomorphic Spaces and Isometric Removal: Exploring the Euclidean Plane

Topology, a fascinating branch of mathematics, deals with the properties of spaces that are preserved under continuous deformations. One intriguing question is whether there exist topological spaces that are homeomorphic to themselves with one point removed. This article delves into such spaces, specifically focusing on the Euclidean plane and its remarkable properties. Additionally, we will introduce the concept of isometry and demonstrate its application in a surprising and elegant construction.

Example: The Circle

One classic example of a space that is homeomorphic to itself with one point removed is the circle, denoted as ( S^1 ).

Circle ( S^1 )

Consider the circle which can be represented as the set of points in the plane satisfying ( x^2 y^2 1 ). If you remove a point, say the point at the top ((0, 1)), the resulting space is homeomorphic to the open interval ((0, 1)) or the real line ( mathbb{R} ). This can be shown by explicitly constructing a homeomorphism. For instance, by wrapping the circle around the real line and mapping the removed point to infinity, we can establish a continuous and bijective function with a continuous inverse.

General Case: Infinite Discrete and Indiscrete Spaces

More generally, any infinite set, such as ( mathbb{R} ) or ( mathbb{Z} ), is also homeomorphic to itself minus a point. Removing any point from ( mathbb{R} ) still leaves it homeomorphic to ( mathbb{R} ) because a homeomorphism can be created by translating the removed point to infinity, resulting in a continuous bijection with a continuous inverse.

Stricter Requirements: The Euclidean Plane

Now, let's consider a more intricate case where the topological space remains not just homeomorphic but isometric after removing a point. A Euclidean plane ( mathbb{R}^2 ) with a specific set of points removed can serve as such an example. Specifically, we remove the points ( (0, 0), (1, 0), (2, 0), (3, 0), ldots ).

The Removal Process

To construct a space that is isometric to itself with one point removed, consider the following transformation:

Define a map ( f: mathbb{R}^2 setminus S rightarrow mathbb{R}^2 setminus S' ) where ( S {(0, 0), (1, 0), (2, 0), (3, 0), ldots} ) and ( S' (0, 0) cup (1, 0) cup (2, 0) cup (3, 0) cup ldots setminus (0, 0) ). This map can be defined as:

[ f(x, y) (x - 1, y) ]

This map is not only continuous but also smooth and an isometry. It preserves distances, meaning that the distance between any two points in ( mathbb{R}^2 setminus S ) is the same as the distance between their images in ( mathbb{R}^2 setminus S' ).

Conclusion

In summary, while the circle ( S^1 ) and any infinite set are homeomorphic to themselves with one point removed, a more stringent requirement involves constructing a topological space that is isometric to itself with one point removed. The Euclidean plane ( mathbb{R}^2 ) with specific points removed, and an isometric transformation, provides a remarkably simple yet elegant solution to this fascinating problem in topology.