Definition (Nowhere dense set) A set $A$ in a topological space $X$ is nowhere dense if the complement of its closure is dense in $X$, i.e. $\overline{(\bar{A})^{\rm c}} = X$.

Definition (Set of first category) A set $A$ in a topological space $X$ is of first category if it is the union of a countable collection of nowhere dense sets, i.e. $A = \bigcup_n E_n$, where each $E_n$ is nowhere dense in $X$.

Baire Category Theorem Let $X$ be a complete metric space. Then no nonempty open subset of $X$ is of first category.

It can be seen that Baire Category Theorem starts from a complete metric space and arrives at a pure topological property. During the proof, the metric property is thrown away after being used, which seems like the scaffold is disassembled after the construction of a building is finished. At the first glance, this is rather weired. However, after some consideration, it is easy to accept this fact on the emotional level in the following way.

In mathematical analysis, we have been familiar with the theoretical framework built up according to this route:

1. A topological space $X$ is based on the definition of openness and open sets. The collection of all open sets $\mathcal{T}$ is called the topology of $X$. At this moment, any element $x$ in $X$ is just an abstract point without the concept of distance between any two of them.
2. A metric space $X$ is based on the definition of a metric function $\rho(x, y)$ on the domain $X \times X$, which can be considered as a measure of the distance between any two points $x$ and $y$. A topology $\mathcal{T}$ can be induced from this metric using open ball $B(x, r)=\{y | \rho(x, y) < r\}$ as its topological basis. Therefore, a metric space is also a topological space but with an additional definition of distance.
3. A normed space $X$ is based on the definition of a norm function $\norm{x}$ on the domain $X$. Now, the space $X$ is assigned the structure required by a linear space. Any element $x$ in $X$ had better be considered as a vector instead of a point, so that the length of this vector is represented by $\norm{x}$. But still, we have no definition about the vector’s direction yet. A metric function $\rho(x, y)=\norm{x - y}$ can be induced from this norm function. Hence, a normed space is a special metric space which has the linear space structure and the measure of a vector’s length.
4. An inner product space $X$ is based on the definition of an inner product function $\langle x, y \rangle$ on the domain $X \times X$, which implicitly holds the information about the relative angle between any two vectors $x$ and $y$. Now, with all the features about distance between two points, length of a vector and angle between two vectors at hand, the inner product space $X$ seems like the Euclidean space that we get used to. A norm function $\norm{x} = \langle x, x \rangle^{\frac{1}{2}}$ can then be induced from the inner product function. Therefore, a inner product space is a special normed space which additionally defines the concept of angle.

The above layer-by-layer construction process leaves us a “clear” impression that a higher layer concept is generated by adding more features on top of the lower layer concepts but without producing any influence on them. However, this is just an illusion! Baire Category theorem tells us that by assigning a metric to a topological space, and then making it complete with respect to this metric, a constraint is imposed on the topological property of $X$, from which no nonempty open subset of $X$ is of first category. This “perturbation” from a higher level concept to a lower one is hidden beneath the elegant theoretical framework, which should not be taken for granted and requires our mathematical insight to discover.