How to find an open cover of X which does not admit a finite subcover?

Question

Let X be an infinite set. Let X be endowed with discrete metric. Give an open cover of X which does not admit a finite subcover.

The open sets in X are singletone sets I know that and X=$\cup_x$ {x} where x$\in$X. But now i dont know how to solve this problem.

Answer 1

It seems like you've done pretty much all of the work already. Each $\{x\}$ is open, so $\mathcal{U} = \{\{x\}\mid x \in X\}$ is an open cover of $X$. For any finite subset $\mathcal{V}$ of $\mathcal{U}$, we have that $\cup_{\{x\} \in \mathcal{V}}\{x\}$ is a subset of $X$ containing finitely many points (since $\mathcal{V}$ is finite), hence it can never be all of $X$ (since $X$ is infinite). Hence $\mathcal{U}$ has no finite subcover.

Answer 2

Consider the cover $\lbrace x \rbrace _{x \in X}$. Clearly, there cannot be a finite subcover.