Metric space Cat. I and Cat. 2

Question

In Munroe1956, Introduction to Measure Theory and Integration, I found this exercise

Let $\Omega$ be a complete metric space. Let $\mu^*(E)=0$ if $E$ is of Cat. I, $\mu^*(E)=1$ if E is of Cat. II. Show that $\mu^*$ is an outer measure, and determine the class of measurable sets.

Are the Cat. I and Cat. II connected with the CAT(k) space concept or they mean something else?

Answer 1

No, Baire category has nothing to do with $\text{CAT}(k)$.

Baire's terminology Category I , Category II is from 1899.

Notation $\text{CAT}(k)$ is from 1987.

Alternate terminology from Bourbaki: nowhere dense = rare, Category I = meagre, Category II = nonmeagre.