Determine the closure $\overline{Y}$

Question

Consider $\mathbb{R}$ with the standard topology, and $Y=[0,\ 1)\subset\mathbb{R}$ with the subspace topology. Determine the closure $\overline{Y}$.

My attempt:

$[0,\ 1]$ is a closed set containing $Y$. So, $\overline{Y}\subset[0,\ 1]$.

$\overline{Y}=Y\cup Y'\supset Y$

i.e., $\overline{Y}\supset[0,\ 1)$

Thus, $[0,\ 1)\subset\overline{Y}\subset[0,\ 1]$

How do I proceed?

Answer 1

You already know that

$$[0,1]\text{ is closed}$$ $$[0,1)=Y\subseteq \overline{Y}\subseteq [0,1]$$

So what is the difference between $[0,1)$ and $[0,1]$? It's one point, namely $\{1\}$. So you have two possible cases:

1. $[0,1)$ is closed and thus $Y=\overline{Y}$
2. $[0,1)$ is not closed and thus $\overline{Y}$ is bigger then $Y$. But since $\overline{Y}\subseteq [0,1]$ and $Y$ and $[0,1]$ differ by only one point it follows that $\overline{Y}$ has to be equal to $[0,1]$

Case 1. is impossible. Indeed, the sequence $1-\frac{1}{n}$ is convergent to $1$ and fully contained in $Y$. Yet $1\not\in Y$. This proves that $Y$ is not closed and thus 1. is not possible.

It follows that the only possibility is 2. $\Box$