Distance from a point to a set

Definition

Let $(X,d)$ be a metric space and let $A\in\mathcal{P}(X)$ be given. For any point $x\in X$ we define the distance between $x$ and $A$ to be:

• $$d(x,A):\eq\mathop{\text{Inf} }_{a\in A}\Big(d(x,a)\Big)$$

We immediately see the following claims:

• Claim 1: if $x\in A$ also then $d(x,A)\eq 0$

Properties

• If $A$ is a closed set in the topology induced by the metric then $d(x,A)\eq 0\iff x\in A$ - Claim 2
• For $x,y\in X$ we see $\Big\vert d(x,A)-d(y,A)\Big\vert\le d(x,y)$ - Claim 3
• For $A\in\mathcal{P}(X)$ define the map: $g_A:X\rightarrow\mathbb{R}$ by $g_A:x\mapsto d(x,A)$ then this map is uniformly continuous - Claim 4

Proof of claims

Grade: C

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Easy stuff, found in page 34-35 of reference

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