Finding topologies of sets in complex space

In summary: Is it true?No, X2 and X3 are not homeomorphic. X2 is a disk, while X3 is an unbounded set. Homeomorphic sets must have the same dimension and same boundary properties.
  • #1
complexnumber
62
0

Homework Statement



Consider the following subsets of [tex]\mathbb{C}[/tex], whose
descriptions are given in polar coordinates. (Take [tex]r \geq 0[/tex] in
this question.)
[tex]
\begin{align*}
X_1 =& \{ (r,\theta) | r = 1 \} \\
X_2 =& \{ (r,\theta) | r < 1 \} \\
X_3 =& \{ (r,\theta) | 0 < \theta < \pi, r > 0 \} \\
X_4 =& \{ (r,\theta) | r = \cos 2\theta \}
\end{align*}
[/tex]
Give each set the usual topology inherited from [tex]\mathcal{C}[/tex].
Which, if any, of these sets are homeomorphic?

Homework Equations





The Attempt at a Solution



[tex]\tau_1 = \varnothing[/tex]. [tex]\tau_2 = \{ B(z,r') \cap X_2 | r'
> 0 \}[/tex]. [tex]\tau_3 = \{ B(z,r') \cap X_3 | r' > 0 \}[/tex]. [tex]\tau_4 =
\varnothing[/tex].

[tex]X_2[/tex] is homeomorphic.

Are my answers correct? I am not sure if the topologies I wrote make sense at all.
 
Physics news on Phys.org
  • #2
Do you even understand what the words you are using mean?

It makes no sense to say that "[itex]X_2[/itex] is homeomorphic". The word "homeomorphic" does not apply to a single set. Two sets are "homeomorphic" is one can be continuously changed into the other. That is, we can shrink, stretch, or otherwise deform a set but we cannot cut, tear, or otherwise break it. In particular, you cannot change the dimension of a set and you cannot change the number of times it intersects itself. While you can change the size of a set with a "continuous" transformation, you cannot change whether it is bounded or not.

Do you mean [itex]\tau_1, \tau_2, \tau_3, \tau_4[/itex] to be the topologies on each of these sets? If so, [itex]\tau_1[/itex] and [itex]\tau_4[/itex] are certainly incorrect! A topology must include the entire set and the empty set at least so a toplogy is never empty. In any case, it is not necessary to write out the topologies- they are just the usual topology on [itex]R^2[/itex] restricted to the given set.

[itex]X_1[/itex] is the circle with center at (0, 0) and radius 1. It is a one dimensional closed path.

[itex]X_2[/itex] is the disk with center at (0, 0) and radius 1. it is a two dimensional, bounded, set.

[itex]X_3[/itex] is the upper half plane. It is two dimsensional and unbounded.

[itex]X_4[/itex] is a "four petal rose". It is one dimensional and intersects itself.

Which, if any, of those sets are homeomorphic to each other?
 
  • #3
HallsofIvy said:
Do you even understand what the words you are using mean?

It makes no sense to say that "[itex]X_2[/itex] is homeomorphic". The word "homeomorphic" does not apply to a single set. Two sets are "homeomorphic" is one can be continuously changed into the other. That is, we can shrink, stretch, or otherwise deform a set but we cannot cut, tear, or otherwise break it. In particular, you cannot change the dimension of a set and you cannot change the number of times it intersects itself. While you can change the size of a set with a "continuous" transformation, you cannot change whether it is bounded or not.

Do you mean [itex]\tau_1, \tau_2, \tau_3, \tau_4[/itex] to be the topologies on each of these sets? If so, [itex]\tau_1[/itex] and [itex]\tau_4[/itex] are certainly incorrect! A topology must include the entire set and the empty set at least so a toplogy is never empty. In any case, it is not necessary to write out the topologies- they are just the usual topology on [itex]R^2[/itex] restricted to the given set.

[itex]X_1[/itex] is the circle with center at (0, 0) and radius 1. It is a one dimensional closed path.

[itex]X_2[/itex] is the disk with center at (0, 0) and radius 1. it is a two dimensional, bounded, set.

[itex]X_3[/itex] is the upper half plane. It is two dimsensional and unbounded.

[itex]X_4[/itex] is a "four petal rose". It is one dimensional and intersects itself.

Which, if any, of those sets are homeomorphic to each other?

Thank you very much for detailed explanation about homeomorphic sets. I did not know those rules and thought the question is asking which set is homeomorphic to the complex space [tex]\mathbb{C}[/tex].

None of these sets are homeomorphic to each other. Is it correct?
 
  • #4
Yes, that is correct.
 
  • #5
I think X2 and X3 are homeomorphic.
 

What is a topology?

A topology is a mathematical concept that describes the properties of a set. It is used to determine the "closeness" of points in a set and the relationships between these points.

How is a topology defined in complex space?

A topology in complex space is defined using the concept of open sets. An open set is a collection of points in complex space where every point is surrounded by a neighborhood that is also contained within the set.

What are the different types of topologies in complex space?

The most commonly used topologies in complex space include the Euclidean topology, the discrete topology, and the Zariski topology. These topologies have different properties and are used for different purposes in mathematics and physics.

Why is finding topologies of sets in complex space important?

Understanding the topology of a set in complex space is crucial for analyzing and solving problems in various fields, including physics, mathematics, and computer science. It helps in determining the behavior of complex systems and finding solutions to complex problems.

What are some techniques for finding topologies of sets in complex space?

Some common techniques for finding topologies of sets in complex space include using algebraic methods, topological methods, and geometric methods. These techniques involve analyzing the properties of sets and their relationships to determine their topologies.

Similar threads

  • Topology and Analysis
Replies
8
Views
395
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
32
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
Replies
3
Views
245
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
4K
Back
Top