Explaining Connected Component: Real Numbers, Rationals, & More

[rainwyz0706]

I'm not quite clear about this notion. Could anyone explain a little bit for me?
Here is the definition:
Let a be an arbitrary point in X . Then there exists a largest connected subset of X
containing a, i.e. a set Ca such that:
• a ∈ Ca and Ca is connected;
• for any connected subset S of X containing a, S ⊆ Ca .
We call such a set Ca the connected component of X containing a, or simply a connected
component of X .
I can see that the connected components of R, Q etc are the singletons.
What about X = {(x, y) ∈ R2 ; x not equal to y} with the topology induced from R2 ?
Btw, if the interior of a subset A of R is the empty set, what are the different possibilities? Obviously, A could be Q, C, R etc.
Your input is greatly appreciated!

 

[WWGD]

Ca is the largest/maximal connected subset containing a, meaning that if S contains a, and

S contains C, then S is not connected.

"I can see that the connected components of R, Q etc are the singletons. "

Q, as a subset of R, is totally disconnected, so that the singletons q_n are the components. Any larger (than {q_n}, for any n ) subset containing q_n is disconnected.

(R , Std. Metric Topology)is connected, so the maximal connected subset containing
R is R itself.

"What about X = {(x, y) ∈ R2 ; x not equal to y} with the topology induced from R2 ?."

Then check to see if X is disconnected: what is the closure of X1={(x,y) in R^2 : x>y}

union X2={(x,y) in R^2: x<y}?. X is disconnected if ClX1/\X2 is empty,

and so is X1/\ClX2 (/\ is intersection, Cl is closure).

In the use I know, components are closed, since , if a subset A is connected, so is its
closure.


"Btw, if the interior of a subset A of R is the empty set, what are the different possibilities? Obviously, A could be Q, C, R etc."

R does not have empty interior. The Baire Category theorem shows this.

Q has empty interior --the irrationals are dense in the reals.

The Cantor set has empty interior
!

 

[rainwyz0706]

Thank you very much for your reply. I have a much clearer picture in my mind.
Just one more question, to find the connected component for X = {(z, w) ∈ C2 ; z not equal to w} with the topology induced from C2, we still need to check to see if X is disconnected right? But it's hard to find two open and disjoint sets whose union is X. Since z not equal to w only means that the real parts and the complex parts are not equal respectively. How do we treat it then? Again, I really appreciate all your input here.

 

[Tinyboss]

I can't see, right away, a straightforward way to show that $$X=\{(z,w)\in\mathbb{C}^2\mid z\ne w\}$$ can't be separated. However, it's not hard to see that it's path connected, implying that it's connected.

First notice that for any u,v,w in C, there's a path from u to v that avoids w (if this is not obvious, draw a picture). Use that to see that there's a path from (u,v) to (z,w) in X, for any choice of (u,v) and (z,w):

First, there's a path from (u,v) to (z,v) lying entirely in C x v which avoids (v,v), by the lemma above. Then, there's a path from (z,v) to (z,w) lying entirely in z x C which avoids (z,z) by the same reasoning. Concatenate the two and you've shown X is path connected, therefore connected.

 

[blue_raver22]

Sure, I'd be happy to explain a little more about connected components for you. The concept of connected components is a fundamental one in topology, which is a branch of mathematics that studies the properties of spaces and their relationships. In simple terms, a connected component is a subset of a space that is "connected" in some way. Let's break down the definition you provided to better understand it.

First, we have the term "connected subset." In topology, a subset of a space is considered connected if every point in the subset can be connected to every other point in the subset by a continuous path. This means that there are no breaks or gaps in the subset, and all points are "reachable" from each other.

Next, we have the concept of a "largest connected subset." This means that within a larger space, there may be multiple connected subsets, but there is one that is the largest and contains all the other connected subsets. This is important because it helps us identify distinct parts of a space that are connected in some way.

Now, let's look at the example you mentioned. In the real numbers, the connected components are the singletons, meaning each individual point is its own connected component. In the rational numbers, the same holds true since each rational number can be connected to all other rational numbers by a continuous path.

However, in the space X = {(x, y) ∈ R2 ; x not equal to y} with the induced topology from R2, the connected components are not singletons. In fact, the connected components in this space are the lines x = y, meaning that any points on the same line can be connected to each other by a continuous path, but points on different lines cannot be connected.

Finally, for the last part of your question, if the interior of a subset A of R is empty, then the possibilities for A are that it is either the empty set, or it is a set that contains only irrational numbers. This is because the interior of a set is defined as the largest open subset contained within that set, and if there are no open subsets contained within a set, then the set must be either empty or contain only irrational numbers, as you mentioned.

I hope this helps clarify the concept of connected components for you. If you have any other questions, please let me know.