View source for Distance from a point to a set

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==Definition==
Let {{M|(X,d)}} be a [[metric space]] and let {{M|A\in\mathcal{P}(X)}} be given. For any point {{M|x\in X}} we define the ''distance between {{M|x}} and {{M|A}}''{{rFAVIDMH}} to be:
* {{MM|d(x,A):\eq\mathop{\text{Inf} }_{a\in A}\Big(d(x,a)\Big)}}
We immediately see the following claims:
* '''Claim 1: ''' if {{M|x\in A}} also then {{M|d(x,A)\eq 0}}{{rFAVIDMH}}
==Properties==
* If {{M|A}} is a [[closed set]] in the [[topology induced by the metric]] then {{M|d(x,A)\eq 0\iff x\in A}}{{rFAVIDMH}} - '''Claim 2'''
* For {{M|x,y\in X}} we see {{M|\Big\vert d(x,A)-d(y,A)\Big\vert\le d(x,y)}}{{rFAVIDMH}} - '''Claim 3'''
* For {{M|A\in\mathcal{P}(X)}} define the [[map]]: {{M|g_A:X\rightarrow\mathbb{R} }} by {{M|g_A:x\mapsto d(x,A)}} then this map is [[uniformly continuous]]{{rFAVIDMH}} - '''Claim 4'''
==Proof of claims==
{{Requires proof|grade=C|msg=Easy stuff, found in page 34-35 of reference|easy=true}}
==References==
<references/>
{{Definition|Metric Space|Functional Analysis|Analysis}}