Give an example of a topological space $X$, and a connected component which is not open in $X$
I know of the following theorems:
- Each connected component is closed;
- If a topological space has a finite number of connected components, then those components are open in $X$.
Because of the second theorem, I need to be on the lookout for a topological space with a infinite number of connected components. I was thinking of something like this:
$$X = \bigsqcup_{n\in \mathbb{N}} \left[n, \dfrac{3}{2} n\right[$$
But this doesn't seem to work since $\sqcup_{n\geqslant 2} [n, \frac{3}{2}[$ as an union of infinite closed sets seems to be closed again.
The open/closed concept in a relative topology can be mindbending. Could someone help me to find an simple example?
Best Answer
Take $$X=\mathbb{Q}$$ the rationals numbers with the Euclidean topology.