Let $X$ be a Noetherian topological space. We know that $X$ is a union of finitley many irreducible components, i.e. $X=X_1\cup\ldots\cup X_n$, where $X_1,\ldots,X_n$ are the maximal closed irreducible subsets of $X$ (irreducible components of $X$).
- Prove that there exists a unique decompositon of $X$ into finitely many disjoint connected components i.e. $X=Y_1\sqcup\ldots\sqcup Y_m$, where $Y_1,\ldots,Y_m$ are maximal closed connected subsets of $X$. Moreover, every connected component is a union of some irreducible components.
- Prove that if $X$ is an affine algebraic group then every connected component of $X$ is irreducible.