Are These Disks With Holes Homeomorphic?

Introduction

In the realm of topology, homeomorphism is a fundamental concept that deals with the study of spaces that can be transformed into each other through continuous deformations. Two spaces are said to be homeomorphic if there exists a continuous function between them that has a continuous inverse. In this article, we will explore the concept of homeomorphism and examine whether two disks with holes are homeomorphic.

Disk with Hole

A disk with a hole is a topological space that consists of a disk with a smaller disk removed from its interior. Mathematically, we can define a disk with a hole as follows:

Let $D \big( (a,b) , r \big)$ be the disk centered at $(a,b)$ with radius $r$. For $0 < r_1 < r$, let $D \big( (a,b) , r_1 \big)$ be the smaller disk centered at $(a,b)$ with radius $r_1$. Then, the disk with a hole can be defined as:

$$D \big( (a,b) , r \big) \setminus D \big( (a,b) , r_1 \big)$$

This space consists of the disk $D \big( (a,b) , r \big)$ with the smaller disk $D \big( (a,b) , r_1 \big)$ removed from its interior.

Homeomorphism

To determine whether two disks with holes are homeomorphic, we need to find a continuous function between them that has a continuous inverse. In other words, we need to find a homeomorphism between the two spaces.

Let $D \big( (a,b) , r \big) \setminus D \big( (a,b) , r_1 \big)$ and $D \big( (c,d) , s \big) \setminus D \big( (c,d) , s_1 \big)$ be two disks with holes. We need to find a homeomorphism between these two spaces.

Theorem 1

Let $D \big( (a,b) , r \big) \setminus D \big( (a,b) , r_1 \big)$ and $D \big( (c,d) , s \big) \setminus D \big( (c,d) , s_1 \big)$ be two disks with holes. If $r = s$ and $r_1 = s_1$, then the two spaces are homeomorphic.

Proof

Let $f: D \big( (a,b) , r \big) \setminus D \big( (a,b) , r_1 \big) \to D \big( (c,d) , s \big) \setminus D \big( (c,d) , s_1 \big)$ be a homeomorphism. We need to show that $f$ is a homeomorphism between the two spaces.

Since $r = s$ and $r_1 = s_1$, we have:

$$D \big( (a,b) , r \big) \setminus D \big( (a,b) , r_1 \big) = D \big( (c,d) , s \big) \setminus D \big( (c,d) , s_1 \big)$$

This implies that $f$ is a homeomorphism between the two spaces.

Theorem 2

Let $D \big( (a,b) , r \big) \setminus D \big( (a,b) , r_1 \big)$ and $D \big( (c,d) , s \big) \setminus D \big( (c,d) , s_1 \big)$ be two disks with holes. If $r \neq s$ or $r_1 \neq s_1$, then the two spaces are not homeomorphic.

Proof

Let $f: D \big( (a,b) , r \big) \setminus D \big( (a,b) , r_1 \big) \to D \big( (c,d) , s \big) \setminus D \big( (c,d) , s_1 \big)$ be a homeomorphism. We need to show that $f$ is not a homeomorphism between the two spaces.

Since $r \neq s$ or $r_1 \neq s_1$, we have:

$$D \big( (a,b) , r \big) \setminus D \big( (a,b) , r_1 \big) \neq D \big( (c,d) , s \big) \setminus D \big( (c,d) , s_1 \big)$$

This implies that $f$ is not a homeomorphism between the two spaces.

Conclusion

In conclusion, we have shown that two disks with holes are homeomorphic if and only if the radii of the disks are equal and the radii of the holes are equal. If the radii of the disks are not equal or the radii of the holes are not equal, then the two spaces are not homeomorphic.

References

• [1] Munkres, J. R. (2000). Topology. Prentice Hall.
• [2] Lee, J. M. (2003). Introduction to Topological Manifolds. Springer-Verlag.

Further Reading

For further reading on topology and homeomorphism, we recommend the following resources:

• [1] Topology by James Munkres
• [2] Introduction to Topological Manifolds by John M. Lee
• [3] Topology and Geometry by Glen E. Bredon

Glossary

• Homeomorphism: A continuous function between two spaces that has a continuous inverse.
• Disk with hole: A topological space that consists of a disk with a smaller disk removed from its interior.
• Radius: The distance from the center of a disk to its boundary.
• Hole: A smaller disk removed from the interior of a disk.

FAQs

• Q: What is a homeomorphism? A: A homeomorphism is a continuous function between two spaces that has a continuous inverse.
• Q: What is a disk with hole? A: A disk with hole is a topological space that consists of a disk with a smaller disk removed from its interior.
• Q: How do I determine whether two disks with holes are homeomorphic? A: To determine whether two disks with holes are homeomorphic, you need to find a continuous function between them that has a continuous inverse.
  Q&A: Are these disks with holes homeomorphic? =====================================================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about homeomorphism and disks with holes.

Q: What is a homeomorphism?

A: A homeomorphism is a continuous function between two spaces that has a continuous inverse. In other words, it is a function that can be reversed without tearing or gluing the space.

Q: What is a disk with hole?

A: A disk with hole is a topological space that consists of a disk with a smaller disk removed from its interior. It is a space that has a hole in the middle.

Q: How do I determine whether two disks with holes are homeomorphic?

A: To determine whether two disks with holes are homeomorphic, you need to find a continuous function between them that has a continuous inverse. In other words, you need to find a homeomorphism between the two spaces.

Q: What are the conditions for two disks with holes to be homeomorphic?

A: Two disks with holes are homeomorphic if and only if the radii of the disks are equal and the radii of the holes are equal.

Q: What happens if the radii of the disks are not equal or the radii of the holes are not equal?

A: If the radii of the disks are not equal or the radii of the holes are not equal, then the two spaces are not homeomorphic.

Q: Can two disks with holes be homeomorphic if they have different shapes?

A: No, two disks with holes cannot be homeomorphic if they have different shapes. Homeomorphism requires that the spaces have the same topological properties, such as connectedness and compactness.

Q: Can two disks with holes be homeomorphic if they have different sizes?

A: No, two disks with holes cannot be homeomorphic if they have different sizes. Homeomorphism requires that the spaces have the same topological properties, such as connectedness and compactness.

Q: Can two disks with holes be homeomorphic if they have different numbers of holes?

A: No, two disks with holes cannot be homeomorphic if they have different numbers of holes. Homeomorphism requires that the spaces have the same topological properties, such as connectedness and compactness.

Q: Can two disks with holes be homeomorphic if they have different orientations?

A: No, two disks with holes cannot be homeomorphic if they have different orientations. Homeomorphism requires that the spaces have the same topological properties, such as connectedness and compactness.

Q: Can two disks with holes be homeomorphic if they have different boundary conditions?

A: No, two disks with holes cannot be homeomorphic if they have different boundary conditions. Homeomorphism requires that the spaces have the same topological properties, such as connectedness and compactness.

Q: Can two disks with holes be homeomorphic if they have different topological properties?

A: No, two disks with holes cannot be homeomorphic if they have different topological properties. Homeomorphism requires that the spaces have the same topological properties, such as connectedness and compactness.

Conclusion

In conclusion, we have answered some of the most frequently asked questions about homeomorphism and disks with holes. We hope that this article has provided you with a better understanding of the concept of homeomorphism and its applications in topology.

References

• [1] Munkres, J. R. (2000). Topology. Prentice Hall.
• [2] Lee, J. M. (2003). Introduction to Topological Manifolds. Springer-Verlag.

Further Reading

For further reading on topology and homeomorphism, we recommend the following resources:

• [1] Topology by James Munkres
• [2] Introduction to Topological Manifolds by John M. Lee
• [3] Topology and Geometry by Glen E. Bredon

Glossary

• Homeomorphism: A continuous function between two spaces that has a continuous inverse.
• Disk with hole: A topological space that consists of a disk with a smaller disk removed from its interior.
• Radius: The distance from the center of a disk to its boundary.
• Hole: A smaller disk removed from the interior of a disk.

FAQs

• Q: What is a homeomorphism? A: A homeomorphism is a continuous function between two spaces that has a continuous inverse.
• Q: What is a disk with hole? A: A disk with hole is a topological space that consists of a disk with a smaller disk removed from its interior.
• Q: How do I determine whether two disks with holes are homeomorphic? A: To determine whether two disks with holes are homeomorphic, you need to find a continuous function between them that has a continuous inverse.