# Distance from a point to a set

## Definition

Let [ilmath](X,d)[/ilmath] be a metric space and let [ilmath]A\in\mathcal{P}(X)[/ilmath] be given. For any point [ilmath]x\in X[/ilmath] we define the distance between [ilmath]x[/ilmath] and [ilmath]A[/ilmath][1] 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 [ilmath]x\in A[/ilmath] also then [ilmath]d(x,A)\eq 0[/ilmath][1]

## Properties

• If [ilmath]A[/ilmath] is a closed set in the topology induced by the metric then [ilmath]d(x,A)\eq 0\iff x\in A[/ilmath][1] - Claim 2
• For [ilmath]x,y\in X[/ilmath] we see [ilmath]\Big\vert d(x,A)-d(y,A)\Big\vert\le d(x,y)[/ilmath][1] - Claim 3
• For [ilmath]A\in\mathcal{P}(X)[/ilmath] define the map: [ilmath]g_A:X\rightarrow\mathbb{R} [/ilmath] by [ilmath]g_A:x\mapsto d(x,A)[/ilmath] then this map is uniformly continuous[1] - Claim 4