a topological space is the union of its irreducible components

Question

if we define irreducible component of a topological space $X$ as the maximal closed irreducible subset of $X$,prove that we can write $X$ as the union of its irreducible components.

how to approach it?

thanks.

Answer 1

Outline of proof:

1. If $Y$ and $Z$ are irreducible components of $X$, then either $Y=Z$ or $Y\cap Z=\varnothing$.

2. Every point $x\in X$ is contained in some irreducible component.

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n. PROFIT!

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Hint: $n=3$.