Prove that $X\subset Y \Rightarrow \bar{X}\subset \bar{Y}$

Question

EDT. $\bar{X}$ is the closure of $X$.

EDT. The exercise is picked from a chapter which discusses metric spaces, therefore I believe it is implied that $X$ and $Y$ are contained in $M$, metric space.

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I am currently completely stuck on how to prove the statement as formulated in the thread title.

If anyone could give me a hint on how one could solve this problem it would be very appreciated.

Answer 1

$\overline{X} = \bigcap \{A | X \subset A, A \text{ closed}\} \subset \bigcap \{A | Y \subset A, A \text{ closed}\} = \overline{Y} \text{ since X contained in Y }$

Answer 2

By definition, $\overline{X}$ is contained in every closed set containing $X$. In particular, $\overline{Y}$ is a closed set containing $X$. So $\overline{X}$ is contained in $\overline{Y}$.