Metric Spaces of the First Category and Second Category

Definition: Let $(X, d)$ be a metric space. $(X, d)$ is said to be of The First Category if $X$ is equal to the union of a countable collection of nowhere dense subsets of $X$.

Definition: Let $(X, d)$ be a metric space. $(X, d)$ is said to be of The Second Category if $X$ is not of the first category.

Observe that if $(X, d)$ is of the second category, and if:

(1)

\begin{align} \quad X = \bigcup_{n=1}^{\infty} E_n \end{align}

where each $E_n \subseteq X$, then there exists some $n \in \mathbb{N}$ for which $E_n$ is nowhere dense (as all of the sets cannot be nowhere dense). But if $E_n$ is nowhere dense, this means that:

(2)

\begin{align} \quad \mathrm{int} (\bar{E_n}) \neq \emptyset \end{align}

So there exists an $x \in \mathrm{int}(\bar{E_n})$. Furthermore, since the interior of a set is open, there exists an open ball centered at $x$ fully contained in $\bar{E_n}$.