Distance between a point to a set in metric spaces

Question

Let $(X, d)$ be any arbitrary metric space. Now fix $y \in X$ and define $d_y: X \rightarrow \mathbb{R}, d_y(x)= d(x,y)$.

How can I prove that $d_y$ is continuous on $X$? (Given any fixed $y \in X$).

Also, let $A \subseteq X$ and fix $x \in X$, define $\rho(x, A) = \inf \{d(x,a) : a \in A \}$,

Show that the closure of $A$ is $\overline{A}=\{x \in X: \rho(x, A)=0 \}$

Finally, let $A \subseteq X$ be a compact subset of $X$.

Prove that $\rho(x, A) = d(x, a)$ for some $a \in A$.

Please, post only one question in one post. Posting several questions in the same post is discouraged and such questions may be put on hold, see meta. Moreover at least some (probably all) questions from your post have been asked here before. — Martin Sleziak, Jul 26 '17 at 10:00
The third question was asked here: For any point $a$ of a compact subset $S$ of a metric space, prove that there exists a nearest point $ c $ to $ a $. — Martin Sleziak, Jul 26 '17 at 10:01
Some posts about the first question: http://math.stackexchange.com/questions/48850/continuity-of-the-function-x-mapsto-dx-a-on-a-metric-space http://math.stackexchange.com/questions/400487/continuity-of-dx-a http://math.stackexchange.com/questions/1361610/prove-that-f-a-x-d-x-a-is-continuous http://math.stackexchange.com/questions/1695885/prove-that-fx-is-continuous-on-m — Martin Sleziak, Jul 26 '17 at 10:01
For the second one see In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$ and other posts linked there. — Martin Sleziak, Jul 26 '17 at 10:02

score 4 · Answer 1 · answered Jul 26 '17 at 03:28

Problem 1

Let $\varepsilon > 0$ be arbitrary, choose $\delta = \epsilon$, try to show that $$ d(x,z)<\delta\implies \lvert d_y(x) - d_y(z)\rvert < \varepsilon $$ on your own.

Problem 2

Recall that the closure of $A$ is the union of $A$ and its limit points. This means that we need to show that if a point $x$ is a limit point of $A$, i.e. if there exists a sequence $(a_n)_n\subseteq A$ that converges to $x$, then $\rho(x,A) = 0$, and that if $\rho(x,A) = 0$, then we can find a sequence $(a_n)_n\subseteq A$ that converges to $x$.

To show the first direction, show that you can always find some $a\in A$ that is arbitrarily close to $x$ by using the given sequence $(a_n)_n$ that converges to $x$. To show the second, use the definition of infimum to try to show the existence of $a\in A$ that are arbitrarily close to $x$.

Problem 3

In metric spaces, compactness is the same as sequential compactness, meaning that every sequence in $A$ has a convergent subsequence that converges inside $A$. First show that you can construct a sequence $(a_n)_n\subseteq A$ so that $d(x,a_n)$ converges to $\rho(x,A)$. Then, show that this sequence $(a_n)_n$ is convergent (hint: compact spaces are complete). Finally, show that $a\in A$, the limit of $a_n$, satisfies $\rho(x,A) = d(x,a)$.

Good luck!

Thanks, is my attempt at Problem 3 correct as follows? Fix $x \in X$ and define the function $d_x: A \rightarrow \mathbb{R}, d_x(a) = d(x, a)$. By the first problem, this function is continuous on $A$ (for fixed $x \in X)$. So by the Extreme Value Theorem (we can apply it here since $A$ is compact), there exists $a^* \in A$ such that $d_x(a^*) = \inf_{a \in A} d_x(a) = \rho(x, A)$. — TeTs, Jul 26 '17 at 14:44
Sounds good! But are you sure you are allowed to use EVT? Seems like you haven't proven it yet, given the nature of the problems. — Tai, Jul 26 '17 at 16:51

Distance between a point to a set in metric spaces

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