Is a Regular Surface an Open Set if Contained in Another Regular Surface in Differential Geometry?

In the realm of differential geometry, the concept of regular surfaces and open sets plays a pivotal role. This article delves into the intricacies of whether a regular surface that is contained within another regular surface is an open set. Specifically, we consider an example where a 2-D Euclidean space is embedded within a 3-D space to explore the nuances of this question.

Introduction to Regular Surfaces and Open Sets

In differential geometry, a regular surface is a subset of Euclidean space that is locally homeomorphic to an open subset of the Euclidean plane. This means that each point on the surface has a neighborhood that is homeomorphic to a plane. A set is considered open if it does not contain its boundary points. The question at hand boils down to whether a regular surface contained within another regular surface is itself an open set within that surface.

The Example: A 2-D Euclidean Space in a 3-D Euclidean Space

Consider a 2-D Euclidean space embedded in a 3-D Euclidean space. Both spaces are regular surfaces. For simplicity, let's take the 2-D plane as a subset of the 3-D space. The question is whether the 2-D plane, as a subset of the 3-D space, is an open set within itself.

In the context of differential geometry, the embedding of a 2-D plane within a 3-D space can be thought of as an immersion. An immersion is a smooth map with the property that the differential at each point is injective. However, the plane itself, as a 2-D manifold, is not an open set within the 3-D space, because an open set in a 3-D space is a space that does not contain its boundary points. The 2-D plane itself is a boundaryless 2-D surface, but it does not contain the "space" around itself within the 3-D context.

Topological Considerations

To understand the concept better, we need to delve into the topology of these spaces. A 2-D Euclidean space is an open set in itself. However, when considered as a subset of a 3-D space, it is no longer an open set because the definition of an open set is relative to the ambient space. In the 3-D context, the 2-D plane is a subset that does not encompass any "volume" beyond itself, making it, in a sense, closed within the 3-D space.

Differential Geometry and Manifolds

The concept of a regular surface in differential geometry is closely tied to the idea of a differentiable manifold. A manifold is a topological space that locally resembles Euclidean space near each point. In the case of a 2-D Euclidean space embedded in a 3-D space, the 2-D manifold is a proper subset of the 3-D manifold. However, the 2-D manifold itself is a complete and self-contained surface, even if it is not an open set when considered within the larger 3-D context.

Implications and Further Explorations

The implications of this question extend to other geometric and topological studies. For instance, the embedding of surfaces in higher-dimensional spaces is a fundamental concept in differential geometry and topology. Understanding whether a surface is an open set in its ambient space has important ramifications for studying the geometric properties of these surfaces and their embeddings.

Conclusion

To summarize, a 2-D Euclidean space embedded in a 3-D Euclidean space is a regular surface in both contexts. However, whether it is an open set within the 3-D space depends on the definition of an open set in the ambient space. The 2-D space itself remains an open set, but as a subset of the 3-D space, it is not considered an open set. This highlights the importance of relative topology and the context in which the space is considered.

References

Differential Geometry of Curves and Surfaces, Manfredo P. do Carmo Introduction to Smooth Manifolds, John M. Lee Topology, James R. Munkres