Does X always has a finite open cover?
X is a metric space. I think it is false because there is a possibility to have closed neighborhood near a point in X.
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What is the definition of an open cover? For a topological space $X$, an open cover is a collection of open sets $\{V_i\}_{i \in I}$, such that $X = \bigcup_{i \in I} V_i$. So just take the open cover $\{X\}$. |
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Every metric space has an open covering consisting of exactly one open set, namely $X$ itself. |
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