The Baire Category Theorem Review

We will now review some of the recent material regarding the Baire Category theorem.

• On The Cantor Intersection Theorem for Complete Metric Spaces page we looked at the Cantor intersection theorem which states the following. If $X$ is a complete metric space and if $(x_n)_{n=1}^{\infty}$ is a sequence of points in $X$ and $(r_n)_{n=1}^{\infty}$ is a sequence of positive real numbers such that $\displaystyle{\lim_{n \to \infty} r_n = 0}$ and:

(1)

\begin{align} \quad ... \overline{B}(x_{n+1}, r_{n+1}) \subseteq \overline{B}(x_n, r_n) \subseteq ... \subseteq \overline{B} (x_1, r_1) \end{align}

• Then there exists exactly one point $x \in X$ such that:

(2)

\begin{align} \quad \bigcap_{n=1}^{\infty} \overline{B}(x_n, r_n) = \{ x \} \end{align}

• On The Baire Category Theorem for Complete Metric Spaces page we looked at another important theorem called the Baire category theorem which states the following. If $X$ is a complete metric space and if $(D_n)_{n=1}^{\infty}$ is a countable collection of open and dense sets in $X$ then the intersection:

(3)

\begin{align} \quad \bigcap_{n=1}^{\infty} D_n \end{align}

• Is dense in $X$.

• We then looked at a corollary to the Baire category theorem on the Corollary to the Baire Category Theorem for Complete Metric Spaces page. We saw that if $X$ is a complete metric space and if $(F_n)_{n=1}^{\infty}$ is a countable collection of closed and nowhere dense sets in $X$ then the union:

(4)

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

• Has empty interior.