202010230028 Problem 2.4.11

Let $(X,d)$ be a metric space, $a\in X$ and $0<r<r'$ .

Prove that the set

$\{ x\in X:\enspace r<d(x,a)<r' \}$

is open in $(X,d)$ .

Setup.

Understand the definition given to each of the following:

First, what is meant by whether a set is open or not in some metric space?

Definition. (open set) Let $(X,d)$ be a metric space. A set $G\subset X$ is said to be an open set if it is a neighborhood of each of its points. (Equivalently, a set $G\subset X$ is said to be an open set
if for each $x\in G$ , there exists an $r>0$ such that $S_r(x)\subset G$ .)

_{Please refer to pg. 20, Jain and Ahmad’s Metric Spaces.}

Second, what is referred to as a neighborhood of some point(s)?

Definition. (neighborhood) Let $(X,d)$ be a metric space and $x\in X$ . A set $N\subset X$ is said to be a neighborhood (nbd) of $x$
if there exists an open sphere centred at $x$ and contained in $N$ ,
i.e., if $S_r(x)\subset N$ for some $r>0$ .

_{Please refer to pg. 19, Jain and Ahmad’s Metric Spaces.}

Third, what is an open sphere?

Definition. (open sphere) Let $(X,d)$ be a metric space. Let $x\in X$ and $r>0$ be a real number. The open sphere with centre $x$ and radius $r$ , denoted by $S_r(x)$ , the subset of $X$ given by $S_r(x)=\{ y\in X:\enspace d(x,y)<r \}$ N.b. An open sphere is always non-empty since it contains its centre at least.

_{Please refer to pg. 16, Jain and Ahmad’s Metric Spaces.}

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