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In my functional analysis textbook there's the statement:
This is often called the category theorem, for the following reason. Let $S$ be either a complete metric space or a locally compact Hausdorff space. If $\left\{ E_i \right\}$ is a countable collection of nowhere dense subsets of $S$, and if $V_i$ is the complement of $\bar{E}_i$, then each $V_i$ is dense in $S$, and the conclusion of the Baire's theorem is that $\bigcap V_i \neq \emptyset$.Hence $S \neq \bigcup E_i$.
Why is $V_i$ dense?
As requested by @PaulFrost.
Set $V_i = S \setminus \overline{E_i}$ for $i \in I$, where $I$ is a countable set of indices. Recall that, given a topological space $(X, \tau)$ and $B \subseteq X$ one have $$ \overline{B} = X \setminus (X \setminus B)^{\circ}.$$ You want to show that $\overline{V_i} = S.$ We have that $$\overline{V_i} = \overline{S \setminus \overline{E_i}} = S \setminus \left( S \setminus (S \setminus \overline{E_i})\right)^\circ = \mbox{you fill the gap} = S \setminus \emptyset = S.$$
To fill the gap use the fact that $E_i$ is nowhere dense in $S$ and note that $B = X \setminus (X \setminus B)$ for any $B \subseteq X$, where $(X, \tau)$ is a topological space.
For the question in the title, the conclusion of the theorem could have been written simply as $S$ is the second category in itself (link). Informally, Bair categories tell you that the first category sets are "small" or "negligible", second category sets are "big". The terminology of first and second category sets is independent of category theory.
