- #1

rainwyz0706

- 36

- 0

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!