I am currently reading Topology and for guidance I'm following my college lectures and the book by Munkres. I'm still in the beginning so I'm having a difficulty to understand the following :
If $X$ has finitely many connected components then any connected component is open. Which subset in $X$ are open and closed?
Firstly, I know that a component is closed so in this case each component can be expressed as the complement of the union of other components (i.e. closed sets) and hence each component becomes open.
Is there any role of the "connected" part here? I can't seem to find it.
Also my second question is what is meant by Which subset in $X$ are open and closed?. Is it saying that I need to find which subsets are clopen or closed/open?
Can I get some help please?
Best Answer
Yes, you are asked to find all the clopen subsets of a topological space with finitely many (connected) components.
The term "connected component" is basically interchangeable with "component" in this context. It is called "connected" because it is a maximal connected subspace. They are called "connected components" not because they are components that happen to be connected, but because they are components with respect to connectivity. (At least, that's how I always thought about it.)
There are, technically, other ways to decompose a topological space, so maybe the author wanted to be precise. Or maybe they, like me, call those things "connected components" out of habit without much more thought.