Exercise 2.19

Question

\def\ints{\mathbb{Z}} Exercise 19: (a) If $A$ and $B$ are disjoint closed sets in some metric space $X$, prove that they are separated. (b) Prove the same for disjoint open sets. (c) Fix $p\in X$, $\delta>0$, define $A$ to be the set of all $q\in X$ for which $d(p,q)<\delta$, define $B$ similarly, with $>$ in place of $<$. Prove that $A$ and $B$ are separated. (d) Prove that every connected metric space with at least two points is uncountable.

ghostofgarborg · Accepted Answer

\def\ints{\mathbb{Z}}

(a) Assume $A$ and $B$ are closed and disjoint 
    $$ \bar A \cap B = A \cap B = \emptyset $$ 
    $$  A \cap \bar B = A \cap B = \emptyset $$
    so that they are separated.
   
   (b) $A^c$ is closed, so by thm 2.27 (c), $\bar B \subset A^c$, and $A \cap \bar B = \emptyset$. 
   Similarly by interchanging $A$ and $B$.

(c) $A$ and $B$ are open and disjoint sets. They are separated by (b).

(d) Fix a point $x$ of a connected metric space $M$, and assume for contradiction that the set of other points is non-empty and at most countable. 
    Then $ D = \{ d(x, y) \}_{y \neq x} $ is countable, and we can pick $r \in \mathbb R \setminus D$ 
    such that $r > 0$ and such that there is a $ y $ with $d(x,y) > r$. By the choice of $r$, 
    $$ M = \{ y \in M : d(x,y) < r \} \cup \{ y \in M : d(x,y) > r \} $$
    which by (c) would imply that $M$ is disconnected, a contradiction.

Exercise 2.19

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Exercise 2.19

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