The following doubt came to my mind when I was reading a proof that if $K\subset Y\subset X$ and $K$ is compact relative to $X$, then $K$ is compact relative to $Y$. 
$$$$The proof assumes that $K$ is compact relative to $X$ and supposes that $\{V\_\alpha\}$ is a collection of sets open relative to $Y$ such that $K\subset \cup\_\alpha V\_\alpha$

Now this is where I got confused. I cannot understand why $Y$ will always have open subsets of itself such that their union covers $K$. I have also taken an example below to better illustrate the difficulty in understanding that I am facing.$$$$

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> If $K\subset Y$, does it necessarily mean that there exist open subsets $\{G\_\alpha\}$ of $Y$ such that $K\subset \cup\_\alpha G\_\alpha$? In other words, does an open cover of $K$ always exist in $Y$ even if $K=Y$?
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I cannot see how it is always possible for a set $K\subset Y$ where $K=Y$ to have open subsets of $Y$ such that the union of these open subsets contains the set $K$ itself.$$$$
As a particular case, consider the set $K=\{z\in\mathbb{C}: |z|\le 1\}$ and let $Y=K$. Thus $K$ is a subset of $Y$. What are the $\textbf{open subsets }$ $\{G\_\alpha\}$ of Y such that $K\subset \cup\_\alpha G\_\alpha$?
$$$$In essence my doubt arises because I cannot see how the union of $\textbf{open subsets}$ of a set can contain the set itself. $$$$I would be grateful if somebody would help me out. Many thanks in advance!