How Do Metric Space Properties Relate to Continuity?

[trees and plants]

Hello. The questions i make here in this thread are basically about like explanations of topics on metric spaces. We know about compactness, completeness, connectedness, separatedness, total boundedness of metric spaces. I know that continuity of the real line means that it has no gaps. What could we say about the properties of the metric spaces i described above in the spirit of the description of the continuity of the real line? I am not talking about the definition which is an abstraction, i am talking about the application of the definition like above in the real line. Thank you.

 

[member 587159]

The connected and compact subset of the real numbers can be characterised as follows:

Connected subset of the real line = interval
Compact set = closed + bounded subset (Heine Borel theorem).

What is your definition of "separated metric space" or what do you mean when talking about separatedness?

 

[trees and plants]

The connected and compact subset of the real numbers can be characterised as follows:

Connected subset of the real line = interval
Compact set = closed + bounded subset (Heine Borel theorem).

What is your definition of "separated metric space" or what do you mean when talking about separatedness?

Thank you. I am sorry i made mistakes. The correct are separable and separability not what i wrote. What about completeness( if i remember correctly it is about sequences, cauchy sequences, convergence)

 

[member 587159]

Thank you. I am sorry i made mistakes. The correct are separable and separability not what i wrote. What about completeness( if i remember correctly it is about sequences, cauchy sequences, convergence)

Every subset of the reals is separable. A subset of the reals is complete if and only it is closed.

 

[trees and plants]

I have another question but is a little off topic I think. Is C which is the set of complex numbers equipped with the metric that is related to the norm, d(x,y)=llx-yll₂=√((x₁-x₀)²+(y₁-y₂)²), where x=(x₁,x₂), y=(y₁,y₂) a metric space? Is it separable? Is it complete if and only if it is closed? Excuse me if these questions have as answer no.

 

[member 587159]

I have another question but is a little off topic I think. Is C which is the set of complex numbers equipped with the metric that is related to the norm, d(x,y)=llx-yll₂=√((x₁-x₀)²+(y₁-y₂)²), where x=(x₁,x₂), y=(y₁,y₂) a metric space? Is it separable? Is it complete if and only if it is closed? Excuse me if these questions have as answer no.

Yes, it is a metric space. It is separable. Can you think of a countable dense subset? Hint: Use density of $\Bbb{Q}$ in $\Bbb{R}$. It is definitely complete, because $\mathbb{R}$ is complete. Asking that it is closed makes little sense because every topological space is automatically closed in itself.

 

[trees and plants]

Yes, it is a metric space. It is separable. Can you think of a countable dense subset? Hint: Use density of $\Bbb{Q}$ in $\Bbb{R}$. It is definitely complete, because $\mathbb{R}$ is complete. Asking that it is closed makes little sense because every topological space is automatically closed in itself.

Thank you, perhaps the answer is that ℚ² is dense in ℝ²? and because ℝ² is homeomorphic to ℂ then ℚ² is dense in ℂ?

 

[member 587159]

Thank you, perhaps the answer is that ℚ² is dense in ℝ²? and because ℝ² is homeomorphic to ℂ then ℚ² is dense in ℂ?

Yes, that's exactly it!

 

[trees and plants]

Yes, that's exactly it!

Oh, i answered it correctly.