Baire's Theorem

Let ( $M, d$ ) be a metric space and $A, B, C \subset M$. Prove that if $A$ is dense in $B$ and $B$ is dense in $C$, then $A$ is dense in $C$.

Let ( $M, d$ ) be a metric space and $A \subset M$. Prove that $A$ is dense in $M$ if and only if the interior of $A^{C}$ is empty.

Let ( $M, d$ ) be a metric space, let $A, B$ be dense subsets of $M$, and suppose that $A$ is open.

Let ( $M, d$ ) be a complete metric space and $\left\{ G_{n}\right\}_{n=1}^{\infty }$ open dense subsets of $X$. Prove that the intersection $\cap_{n=1}^{\infty } G_{n}$ is dense in $M$.

Let $G$ be an open dense set in a metric space $M$, and let $N$ be a set with empty interior. Prove that $G \backslash N$ is dense in $M$

Let $\mathcal{A}$ be the collection of all nowhere dense subsets of a metric space $(M, d)$. Prove that:

Let $A$ be a closed subset of a metric space ( $M, d$ ). Prove that the boundary of $A$ is nowhere dense. What if $A$ is an open set?

Let $A$ be a dense subset of a metric space. Prove or disprove: the set $A^{C}$ is nowhere dense.

Prove that the graph of $y=x$ is nowhere dense in $\mathbb {R}^{2}$.

Prove that a set is the boundary of some open set if and only if it is closed and nowhere dense.

Let $A$ be a closed subset of a metric space such that every point of $A$ is a cluster point of both $A$ and $A^{C}$. Prove that $A$ is nowhere dense.

Prove that a set $A$ in a metric space is nowhere dense if and only if for every open set $G$, there is an open set $G_{1} \subset G$ such that $G_{1} \cap A=\varnothing$.

Let $E$ be a subset of a complete metric space $M$ and suppose that both $E$ and $E^{C}$ are dense. Prove that at most one of them is the union of a countable collection of closed sets. Conclude that the set of rational numbers in $[0,1]$ is not the intersection of a countable collection of open sets.

Let $M$ be a complete metric space without isolated points and let $\left\{ F_{n}\right\}$ be a countable collection of closed, nowhere dense sets. Prove that $\left(\cup_{n=1}^{\infty } F_{n}\right)^{C}$ is dense in $M$ and uncountable.

Let ( $M, d$ ) be a complete metric space without isolated points. Prove that $M$ is uncountable. Conclude that the set of all real numbers in uncountable.

Prove that the set of rational numbers is not a $G_{\delta }$ set.

Let ( $M, d$ ) be a metric space. Prove that it is of the second category if and only if, whenever $M=\cup_{n=1}^{\infty } F_{n}$ where each $F_{n}$ is closed, at least one of the sets $F_{k}$ contains an open ball.

Prove that if a subset of a metric space is of second category then it must be uncountable. Conclude that the interval $[0,1]$ is uncountable.

Prove that in a complete metric space, a set is of the first category if and only if its complement contains a dense $G_{\delta }$ set.

Prove that if $A \subset \mathbb {R}$ is a set is of first category then $A^{C}+A^{C}=\mathbb {R}$, if $A^{C}+A^{C}=\left\{ x+y: x \in A^{C}, y \in A^{C}\right\}$.

Give an example of a metric space that is of second category but not complete.

Let ( $M, d$ ) be a separable metric space and $E$ a bounded subset of $M$. Prove that $(E, d \mid E)$ is separable.

Prove that every totally bounded metric space is separable.

Prove that a compact metric space is separable.

If $P$ is an infinitely differentiable function such that for each $x$, there is an $n$ with $P^{(n)}(x)=0$, prove that $P$ is a polynomial. (Note $n$ depends on $x$.)

Suppose that $f:[0,+\infty ) \rightarrow [0,+\infty )$ is a continuous function with the following property: for all $x \in [0,+\infty )$, the sequence $f(x), f(2 x), f(3 x), \ldots$ tends to 0 . Prove that $\lim_{t \rightarrow \infty } f(t)=0$.

Let $\left\{ f_{n}\right\}$ be a sequence of real-valued functions on $[0,1]$ defined by $f_{0}=f \in C[0,1]$ and $f_{n}$ is an antiderivative of $f_{n-1}$. Suppose that for each $x \in [0,1]$ there is $n \in \mathbb {N}$ for which $f_{n}(x)=0$. Prove that $f=0$.

Let ( $M, d$ ) be a metric space, let $\left\{ f_{n}\right\}$ be a sequence of continuous, real-valued functions on $M$, and suppose that there is a set $A \subset M$ of second category such that $\sup \left\{ \left|f_{n}(x)\right|: n \in \mathbb {N}\right\}$ is finite for each $x \in A$. Prove that there exists a ball $B\left(x_{0}, r\right)$ and $K>0$ such that $\left|f_{n}(x)\right| \leq K$, for all $n \in \mathbb {N}$ and all $x \in B\left(x_{0}, r\right)$.

Let ( $M, d$ ) be a metric space and let $G$ be an open subset of $M$. Prove that $G$ is of second category in itself.

Let ( $M, d$ ) be a metric space and let $G$ be a $G_{\delta }$ subset of $M$. Prove that $G$ is of the second category in itself.

Let $\left\{ f_{n}\right\} \subset C[0,1]$ that converges to a real-valued function $f$, for all $x \in [0,1]$. Prove that for every $\varepsilon >0$ there is a dense subset $D_{\varepsilon } \subset [0,1]$ such that if $x \in D_{\varepsilon }$, then there is an open interval $I$ containing $x$ and a positive integer $N_{x}$ such that for all $n>N_{x}, \sup \left\{ \left|f_{n}(y)-f(y)\right|: y \in I\right\} \leq \varepsilon$.