It-is-given-a-family-of-open-interval-set-U-r-r-Q-of-R-that-satifies-condition-r-Q-r-U-r-Prove-that-there-exists-a-family-set-U-r-r-Q-which-not-cover-R-or-gt-0-r-Q-U-r-

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Question Number 182241 by Matica last updated on 06/Dec/22

$$\mathrm{It}\mathrm{is}\mathrm{given}\mathrm{a}\mathrm{family}\mathrm{of}\mathrm{open}\mathrm{interval}\mathrm{set}{\left(U_r\right)}_{r\in \mathbb{Q}}\mathrm{of}\mathbb{R}$$$$\mathrm{that}\mathrm{satifies}\mathrm{condition}\mathrm{\forall }r\in \mathbb{Q},r\in U_r.$$$$\mathrm{Prove}\mathrm{that}\mathrm{there}\mathrm{exists}\mathrm{a}\mathrm{family}\mathrm{set}{\left(U_r\right)}_{r\in \mathbb{Q}}\mathrm{which}\mathrm{not}\mathrm{cover}\mathbb{R}$$$$\mathrm{or}\mathrm{\forall }\epsilon >0,\lambda \left(\underset{r\in \mathbb{Q}}{\cup }{U}_r\right)⩽\epsilon .$$

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