Let $X$ be a a connected set of real numbers. If every element of $X$ is irrational then the cardinality of $X$ is
- Infinite
- Countably finite
- $3$
- $1$
Let $(X,d)$ be a metric space and let $A⊆X$. For $x∈X$, define $d(x,A) = \inf\{d(x,a):a∈A\}.$ If $d(x,A)=0$ for all $x∈X$, then which of the following must be true?
- $A$ is compact
- $A$ is closed
- $A$ is dense in $X$
- $A=X$
My thoughts:
For 1st question, 4 is correct as rationals or irrationals are dense.
For 2nd question, 3 is correct as in that case every point of $A$ becomes a limit point of $X$.
Are my conclusions correct?