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The following are a few statements in various metric spaces mcqs that I couldn't figure if they are true or false. Please offer some help to get answer them

Let $(X,d)$ be a metric space

1) If $ A,B \subseteq X $ and $ A,B $ are bounded

$\mathrm{dist}(A,B)>0 \Rightarrow A \bigcap B = \emptyset$

2)$ A \subseteq X $ and $A$ is nowhere dense $\Rightarrow X$\ $\overline A $ is dense

$ \emptyset \neq S \subseteq X $

3) $A \subseteq S$ and $A$ is closed in S $\Rightarrow $ A is closed in $X$

4) $A \subseteq X \Rightarrow \overline {A \bigcap S} $ (closure wrt to S)= $\overline A \bigcap S$ (A's closure wrt to X)

if $ d_1$ and $d_2$ are metrics on X and $ \emptyset \neq A \subseteq X$

5) $d_1(x,y) \le d_2(x,y)$ for each x,y $ \in X \Rightarrow$ G is $ d_2$ open for each $d_1$ open subset G of X

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    $\begingroup$ 1) Did you try the contrapositive? It does not matter whether $A$ and $B$ are bounded. 5) Did you try showing the complement of $G$ is $d_2$ closed? 9) Too many questions in one question. $\endgroup$ – Julien Jul 2 '13 at 5:40
  • $\begingroup$ 1) Yes the contrapositive is true and hence the statement is true right? Just wasn't sure of it $\endgroup$ – user68099 Jul 2 '13 at 5:45
  • $\begingroup$ 3) What is $S$? $\endgroup$ – Hawk Jul 2 '13 at 5:46
  • $\begingroup$ @sidht $ S$ is a subset of X for the questions 3 and 4 $\endgroup$ – user68099 Jul 2 '13 at 5:47
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HINTS:

  1. is completely trivial, but here’s a hint anyway: if $x\in A$, then $\operatorname{dist}(x,A)=0$.

  2. This just requires you to know the definition of nowhere dense. If $A$ is nowhere dense, then $\operatorname{cl}A$ does not contain any non-empty open set.

  3. In the real line, $[1,2)$ is closed in $(0,2)$.

  4. What if $A=(0,1)$ and $S=[1,2]$ in $\Bbb R$?

  5. If $G$ is $d_1$-open and $x\in G$, then there is an $\epsilon>0$ such that $B_{d_1}(x,\epsilon)\subseteq G$. Show that $B_{d_2}(x,\epsilon)\subseteq G$.

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