Baire's Theorem and Applications

The Baire Category Theorem

Lemma 1: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X, \tau)} be a topological space and let $A\subseteq X$. If $A$ is a nowhere dense set then for every $U\in \tau$ there exists a $B\subseteq U$ such that $A\cap {\bar {B}}=\emptyset$.

Theorem 1 (The Baire Category Theorem): Every complete metric space is of the second category.

• Proof: Let $X$ be a complete metric space. Then every Cauchy sequence $(x_{n})_{n=1}^{\infty }$ of elements from $X$ converges in $X$. Suppose that $X$ is of the first category. Then there exists a countable collection of nowhere dense sets $A_{1},A_{2},...\subset X$ such that:

${\begin{aligned}\quad X=\bigcup _{i=1}^{\infty }A_{i}\end{aligned}}$

• Let $U\subset X$. For each nowhere dense set $A_{i}$, $i\in \{1,2,...\}$ there exists a set $B_{i}\subset U$ such that $A_{i}\cap {\bar {B_{i}}}=\emptyset$.

• Let $B_{1}(x_{1},r)\subset U$ be a ball contained in $U$ such that $A_{1}\cap {\bar {B_{1}}}=\emptyset$. Let $B_{2}\left(x_{2},{\frac {r}{2}}\right)\subseteq B_{1}$ be a ball contained in $B_{1}$ whose radius is ${\frac {r}{2}}$ and such that $A_{2}\cap {\bar {B_{2}}}=\emptyset$. Repeat this process. For each $n\in \{2,3,...\}$ let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_n \left (x_n, \frac{r}{n} \right )} be a ball contained in $B_{n-1}$ whose radius is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{r}{n}} and such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_n \cap \bar{B_n} = \emptyset} and such that $A_{n}\cap {\bar {B_{n}}}=\emptyset$.

• The sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_n)_{n=1}^{\infty}} is Cauchy since as $n\in \mathbb {N}$ gets large, the elements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_n} are very close. Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a complete metric space, we must have that this Cauchy sequence therefore converges to some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \in X} , i.e., Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} x_n = p} .

• Now notice that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \bar{B_n}} for all $n\in \mathbb {N}$ because if not, then there exists an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \in \mathbb{N}} such that $x\not \in {\bar {B_{n}}}$ for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \geq m} . Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\bar{B_n})^c} is open and so there exists an open ball Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in B \subset (\bar{B_n})^c} but then $\lim _{x\to \infty }x_{n}\neq x$ because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_m \not \in B} for all $m\geq n$.

• Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \bar{B_n}} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{N}} then since $A_{n}\cap {\bar {B_{n}}}=\emptyset$ we must have that then $x\not \in A_{n}$