Irreducible closed subset

Question

The dimension of a topological space $X$ is defined as the supremum of all integers $n\ge0$ for which there is a strictly increasing chain of $n+1$ irreducible closed subsets of $X$. Thus is the dimension of the empty topological space $-\infty$? Is the dimension $>-\infty$ for all nonempty topological spaces?

Answer 1

Yes and yes. For your first question, since there's no such $n \ge 1$, we have $\sup \emptyset = -\infty$. So you're right. For your second question, it suffices to show that any nonempty space $X$ has at least one irreducible closed subset, so that $\dim X = \sup \{0, \dots\} \ge 0$

Since $X$ is nonempty, we can pick some $x \in X$. By lemma 8.3 (1), the closure of any irreducible subset is irreducible. Since $\{x\}$ is irreducible, $\overline{\{x\}}$ is an irreducible closed subset of $X$. So we're done.

Note: I refer to definition 8.1 (1) for the definition of irreducible subset.