# Which of these statements regarding metric spaces are true?

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

• 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. – Julien Jul 2 '13 at 5:40
• 1) Yes the contrapositive is true and hence the statement is true right? Just wasn't sure of it – user68099 Jul 2 '13 at 5:45
• 3) What is $S$? – Hawk Jul 2 '13 at 5:46
• @sidht $S$ is a subset of X for the questions 3 and 4 – user68099 Jul 2 '13 at 5:47

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 $B_{d_2}(x,\epsilon)\subseteq G$.