Connected component

Question

please how to prove that the connected components form a partition of the topological space ?

I know that they are closed and disjoint but how to conclude?

Thank you

Answer 1

There is a relation $\sim$ on the elements of the topological space $X$ prescribed by:$$x\sim y\iff\text{ a connected set }C\subseteq X\text{ exists with }x,y\in C$$ This relation can be shown to be an equivalence relation, so induces a partition on $X$.

Also it can be shown that the equivalence classes are the components of $X$.

So with that you are ready.

If $[x]$ denotes the equivalence class represented by $x$ then $[x]$ can be shown to be the union of all connected subsets of $X$ that contain $x$ as element, and can be recognized as the component that has $x$ has element.