Baire Category Theorem and Applications

This post was inspired by a talk given by Professor David Damanik from Rice University. In this post, we will prove the Baire Category theorem for complete metric spaces and briefly talk about applications.

Notation. Let $X$ be a metric space and $v \in X, r \in \mathbb R^+$ . Then, $B(r, v)$ denotes the open ball centered at $v$ with radius $r$ .

Definition. A space $X$ is Baire if and only if the countable intersection of dense open sets in $X$ is also dense.

For example, the space $[0, 1]$ is Baire. Consider two dense open sets $(0, \frac 1 2)\cup (0, \frac 1 2)$ and $(0, \frac 1 3 ) \cup (\frac 1 3, 1)$ . The intersection of the two is $[0, 1] \setminus \{0, \frac 1 3, \frac 1 2\}$ which is also dense in $[0, 1]$ .

Theorem. Any complete metric space is a Baire Space.

proof. We prove by taking a sequence of balls that form an inclusive chain. Call the space $X$ and let $(O_k)_{k=1}^\infty$ be a sequence of dense open sets. Recall that a subset is dense if and only if every nonempty open subset intersects with the subset. Therefore, take an arbitrary open set $V \subseteq X$ and show that it intersects with each $O_k$ at a common point for every $k \geq 1$ . The two sets $O_1, V$ has a nonempty open intersection. Take a closed ball $\overline B(v_1, r_1)$ included in the intersect. Repeat the procedure with the open ball $B(v_1, r_1)$ and $O_2$ to obtain $v_2, r_2$ , and so on with $B(v_k, r_k)$ and $O_{k + 1}$ to obtain $v_{k + 1}, r_{k+1}$ .

Clearly, the sequence of closed balls form an inclusive chain.

$\overline B(v_1, r_1) \supseteq \overline B(v_2, r_2) \supseteq \cdots \supseteq \overline B(v_k, r_k) \subseteq \cdots$

The space is complete, so $v_k \rightarrow v_\infty$ for some $v_\infty \in X$ as $k$ goes to infinity. By construction, $v_k$ is common to all of the closed balls, therefore to all the sets $O_k$ and the initial open set $V$ . □

Lets reformulate the result. A $G_\delta$ set denotes a countable intersection of open sets. The Baire category theorem says that if the component sets of the open sets are dense, then the resulting $G_\delta$ set is also dense.

The Baire category theorem can be used to study decomposition of operators. A Schrodinger operator decomposes the space into eigenstates that correspond to pure point, singular continuous, absolutely continuous spectra. The Wonderland theorem (by Barry Simon) proves that the eigenstates corresponding to pure point and absolute continuous spectra form a dense subset in the space of all states. The proof involves the Baire category theorem.

References and further reading:

https://en.wikipedia.org/wiki/Baire_category_theorem

http://www.math.caltech.edu/SimonPapers/234.pdf

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