I had just started studying Baire's Category Theorem and I was at first having problem with the seemingly technical definitions of nowhere dense sets,first category and second category sets.But slowly it was revealed to me that these definitions are not coming out of the blue,they have some significance.The idea of nowhere dense is quite simple to interpret,it is nowhere dense in the sense that it is not dense in any non-empty open set.But first category and second category sets are a bit complicated to be appreciated at first glance.Sometimes we say that first category sets are 'meagre' or small in the sense that they are countable(small enough) union of nowhere dense(sparse) sets.That would seem quite convincing but at the next glance you would figure out something else.Suppose we consider the metric space $\mathbb Q$ with usual distance $|.|$,consider an enumeration $(r_n)$ of this set.Then $\mathbb Q=\large\cup_{n\in \mathbb N}$$\{ r_n\}$,where each of the singletons in nowhere dense as $\mathbb Q$ has no isolated points.So,$\mathbb Q$ is of first category,but notice that here it is not small in that sense because it is the entire space.So,the thing small is not understood properly here.Now,it is best understood when we work with complete metric space.We know that complete metric spaces are of second category.Now here if we have a first category set,then it is clear that it cannot contain any open non-empty set(By Baire's Category theorem):
We can proceed directly that if $X$ is a complete metric space and $U\subset A$,$U$ being a non-empty open set,then $U$ is of second category.So any superset of $U$ in particular $A$ must be of second category.So,a first category set in a complete metric space cannot contain a non-empty open set.
So,a first category set is indeed 'meagre' in true sense of the term when we consider a complete metric space.Also,we can say that first category set is 'meagre' because in complete metric space,its complement is of second category.
So,I think the terms 'meagre' or 'small' in terms of category is best understood and in fact justified when we look at complete metric spaces.In fact,Baire developed his theorem for $\mathbb R$ which is a complete metric space.So,it think he thought of these terminologies like 'meagre'.Is it correct?