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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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2 Answers 2

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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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Thank you for answering. Can you please explain more? so, it is true that X always has a finite open cover? –  ANOVA Nov 8 '12 at 5:00
    
I don't know what to explain, really! –  Mariano Suárez-Alvarez Nov 8 '12 at 5:04
2  
{X} is a finite open cover of X. –  The Substitute Nov 8 '12 at 5:04

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