Let $M$ be a {as||complete} metric space and let $\{C_n{\}}_n$ be a {as||collection of closed subsets} of $M$ such that $$M = \bigcup_{n \in N}C_{n}$$Then {ha||at least one} of the $C_n$ {ha||contain an open ball}
{ha||$$B(x,r) = { y \in M : d(x,y) <r }$$}
ie, {ha||at least one} $C_n$ has {ha||an interior point}.

A set that does not contain an interior point is called nowhere dense. Sometimes when we apply this theorem, we do not need $C_n$ to be closed. Then the result is that their closures must contain and open ball. ie, we cannot have all $C_n$ be all nowhere dense.

We can use this theorem to prove that there exists a continuous function that is nowhere differentiable

Suppose BWOC that there is some collection of closed subsets of $M$ such that $\bigcup _n{C}_n=M$ and all $C_n$ are nowhere dense.

Since $M$ contains an open ball but $C_1$ does not, we have $M\ne C_1$. Thus there is some $p_1\in M\setminus C_1$. Since $C_1$ is closed but $M\setminus C_1$ is open, this implies that $\mathrm{\exists }{ϵ}_1>0$ s.t. $B(p_1,{ϵ}_1)\cap C_1=\varnothing$.

Now, $B(p_1,\frac{{ϵ}_1}{3})\not\subset C_2$ means there exists some $p_2\in M\setminus C_2$. And since $C_2$ is closed, there exists $0<{ϵ}_2<\frac{{ϵ}_1}{3}$ such that $B(p_2,{ϵ}_2)\cap C_2=\varnothing$

Now, suppose there we have found $k$ such points. Then for each $j=1,\dots ,k$ we have $p_j\not\subset C_j$ with${ϵ}_j$ such that $B(p_j,{ϵ}_j)\cap C_j=\varnothing$

Since $B(p_k,\frac{{ϵ}_k}{3})\not\subset C_{k+1}$, there exists some $p_{k+1}\in B(p_k,\frac{{ϵ}_k}{3})$ such that $p_{k+1}\notin C_{k+1}$

Then there exists ${ϵ}_{k+1}<\frac{{ϵ}_k}{3}$ such that $B(p_{k+1},{ϵ}_{k+1})\cap C_{k+1}=\varnothing$

Thus, by indiction, we have found a sequence of points $\{p_k{\}}_k\subset M$ and ${ϵ}_k\in (0,{ϵ}_1)$ such that for all $k$,

1. $p_j\in B(p_{j-1},\frac{{ϵ}_{j-1}}{3})$
2. $B(p_j,{ϵ}_j)\cap C_j=\varnothing$

This sequence $\{p_k{\}}_k$ is Cauchy because for all $k,\ell \in \mathbb{N}$, we have

Since $M$ is complete, there exists some $p\in M$ such that $p_k\to p$ as $k\to \mathrm{\infty }$.

Now, for all $k\in \mathbb{N}$, we have

So as $\ell \to \mathrm{\infty }$, we have

ie, $p\in B(p_k,{ϵ}_k)=B_k$ for all $k$. And each of these balls has $B_k\cap C_k=\varnothing$. Thus $p$ is not in any of the $C_k$ ie $p\notin \bigcup _k{C}_k=M$