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Supose $X$ is a metric space and let $f:\mathbb{R}\rightarrow X$ continuous and Surjective function (onto) show that $X$ is separable.

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    I mean.. this is a very basic exercise. Did you actually give any effort? –  Oct 10 '20 at 16:10
  • i am anundergraduate student i dont know how to solve it, could you help ? – 領域展開 Oct 10 '20 at 16:22
  • i dont know how this comment add something usefull to the post and got an upvote – 領域展開 Oct 10 '20 at 16:45
  • What is your definition of Separable? –  Oct 10 '20 at 19:12
  • let $(X,d)$ be a metric space if it exist a $Y\subseteq X $ where $Y$ is countable and $\bar{Y}=X $ then $X$ is separable – 領域展開 Oct 10 '20 at 23:09
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    @PetrosK This comment got an upvote as it is consent in this community that people should show what they have tried :) People are usually quick to vote to close questions otherwise. This is nothing personal, many people just feel that the aim of this site is to discuss interesting math and not to solve homework problems for others. – Severin Schraven Oct 11 '20 at 10:53

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Hints:

$1)$ $\mathbb R$ is separable: $\overline{\mathbb Q} = \mathbb R$.

$2)$ Let $X$ and $Y$ be any two metric spaces and $A$ be any subset of $X$. If $f : X \to Y$ is continuous, then $f(\overline{A}) \subset \overline{f(A)}$.

$3)$ Let $M$ and $N$ be any two sets. If $f: M \to N$ is any mapping and $A \subset M$ is countable, then $f(A)$ is countable.