Showing Connectedness: Other Ways Than Path Connectedness

[pivoxa15]

Homework Statement

What are some ways of showing two sets are connected apart from showing they are path connected?

To show path connectedness is it okay if I nonrigorously draw a path continously on a page that touches every point in the boundary of the sets? i.e two closed circles are path connected but finding a curve can be difficult in some situations.

The Attempt at a Solution

Another way to show connectedness is to show there is a set shared in each of the sets. Any other ways?

 

[StatusX]

What are some ways of showing two sets are connected apart from showing they are path connected?

Connectedness is a property of a single set. Are you asking how to show the union of two sets is connected? One way is if the sets are themselves (path) connected, and they intersect, then their union is (path) connected.

To show path connectedness is it okay if I nonrigorously draw a path continously on a page that touches every point in the boundary of the sets?

I don't see what you mean. Are you saying you're drawing a path in the set that meets every point in the boundary of the set (so that in particular the set contains its boundary and so is closed)? No, this is neither necessary nor suffcient for the set to be connected.

i.e two closed circles are path connected but finding a curve can be difficult in some situations.

The union of two circles is path connected iff they intersect.

 

[pivoxa15]

I don't see what you mean. Are you saying you're drawing a path in the set that meets every point in the boundary of the set (so that in particular the set contains its boundary and so is closed)? No, this is neither necessary nor suffcient for the set to be connected.

How about being able to connect any two points in the set via a path? But show it non rigorously without any maths. Just use intuition. This is given offcourse that the set looks obvious like the union of two closed circles that intersesct.

 

[quasar987]

You are actually asking if it's OK to show connectedness non-rigorously. But OK for what? It's relative. It's OK if you're working on some problem and you want to use the fact that some space is (obviously) connected to prove a more interesting result. But it's probably not OK in an exam as the answer to "Show such and such space is connected." As for showing connectedness of a space, you can also try a proof by contradiction. Suppose it's not connected, then.. blahblah. ==><==

Also, if a topological space is connected, then its number of connected components is 1. So you can use the fact that the number of connected components is a topological invariant (meaning if two spaces are homeomorphic, then they share the same number of connected components). This means that if you show that the space is homeomorphic to some other space whose number of connected components is 1, then the original space is connected. If the number of connected components of the second space is not 1, then the original space is not connected.

 

[StatusX]

Another way in the spirit of your idea is to use the face that the image of a (path) connected space is (path) connected. So if you can find a surjective function from a (path) connnected space X (say, the unit interval) to a space Y, then Y is (path) connected.