Relations between compactness and connectedness

[soTo]

Hello there,

This might be probably a simple question, but my wondering was:

Is there any relation between the compactness and the connectedness of a topological space?

Let us consider the specific example (of interest for me) of a subdomain D of a 3D Riemannian manifold.

i) If D is compact, can I say that it is necessarily connected?

ii) If D is simply-connected, can I say that it is compact?

iii) If D is multi-connected, can I say that it is compact?

If one or another answer to these questions is negative, can you please provide me with an example?

Regards

 

[HallsofIvy]

Hello there,

This might be probably a simple question, but my wondering was:

Is there any relation between the compactness and the connectedness of a topological space?

Let us consider the specific example (of interest for me) of a subdomain D of a 3D Riemannian manifold.

i) If D is compact, can I say that it is necessarily connected?

Let D be the ball of radius 1 with center at (-1, 0, 0) union the ball with radius 1 and center at (1, 0, 0) in R³. That set is compact but not connected.

ii) If D is simply-connected, can I say that it is compact?

The line, R¹ is simply connected but not compact.

iii) If D is multi-connected, can I say that it is compact?

I don't know what you mean by "multi- connected"- has multiple components?
If so, the union of $[1, \infty)$ and $(-\infty, -1]$ has two components but is not compact. The set of all integers, as a subset of the real line, has an infinite number of components but is not compact.

If one or another answer to these questions is negative, can you please provide me with an example?

Regards

The answer to all of those questions is negative. I really have no idea why you would think that "connected" and "compact" are related. They both start with "c" is about as close as you will get!

 

[soTo]

Thanks a lot for your answer and your examples HallsofIvy!

 

[mathwonk]

i'm going to go out on a limb here and maybe say something false, in an attempt to show that the three concepts are quite different.

1) a manifold M is connected if every continuous map from {0,1}-->M extends to a continuous map [0,1]-->M.

2) a manifold M is simply connected if every continuous map circle-->M

extends to a continuous map disc-->M.

3) a manifold M is compact if there is a surjective continuous map [0,1]-->M.

so you see these three properties are pretty dissimilar.

 

[blue_raver22]

,



Dear [Name],

Thank you for your question regarding the relations between compactness and connectedness in a topological space, specifically in the context of a subdomain D of a 3D Riemannian manifold. These concepts are fundamental in topology and understanding their relationship is important for many applications in mathematics and science.

To answer your first question, yes, if a topological space is compact, then it is necessarily connected. This is a well-known result in topology known as the "Heine-Borel theorem". Essentially, compactness implies that every open cover of the space has a finite subcover, and this finite subcover must cover the entire space, thus making it connected.

However, the reverse is not always true. A connected topological space may not necessarily be compact. For your second and third questions, if a topological space is simply-connected or multi-connected, it does not necessarily mean that it is compact. A simple counterexample for this is the open unit disk in the complex plane. It is simply-connected and multi-connected, but it is not compact.

I hope this answers your questions. If you have any further inquiries, please do not hesitate to reach out.

Best regards,