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Question Number 204702 by mnjuly1970 last updated on 25/Feb/24

$$$$$$\mathrm{prove}\mathrm{that}:$$$$\mathrm{cl}\left(\mathbb{Q}\times \mathbb{Q}\right)\stackrel{?}{=}{\mathbb{R}}^2$$$$note:(X,d)isametricspace$$$$,A\subseteq X:x\in \stackrel{-}{A}=\mathrm{cl}\left(A\right)\iff \mathrm{\forall }r>0,B_r\left(x\right)\cap A\ne \varphi$$

Answered by witcher3 last updated on 25/Feb/24

$$\stackrel{-}{\mathrm{Q}}=\mathbb{R}$$$$\mathrm{cl}(\mathrm{Q}\ast \mathrm{Q})\subset \mathrm{Cl}(\stackrel{-}{\mathrm{Q}}\ast \stackrel{-}{\mathrm{Q}})={\mathbb{R}}^2$$$$\mathrm{let}\mathrm{x}(\mathrm{a},\mathrm{b})\in {\mathbb{R}}^2=(\mathrm{x}$$$$\Rightarrow \mathrm{\exists }{\mathrm{U}}_{\mathrm{n}},{\mathrm{V}}_{\mathrm{n}}\in \mathrm{Q}$$$${\mathrm{u}}_{\mathrm{n}}\to \mathrm{a};{\mathrm{v}}_{\mathrm{n}}\to \mathrm{b}$$$$\Rightarrow \mathrm{\forall }ϵ>0\mathrm{\exists }\mathrm{N};\mathrm{\forall }\mathrm{n}⩾\mathrm{N};\mid {\mathrm{U}}_{\mathrm{n}}-\mathrm{a}\mid <ϵ;\mathrm{\forall }{ϵ}^{\prime }>0\mathrm{\exists }{\mathrm{N}}^{\prime }\in \mathbb{N};\mathrm{\forall }\mathrm{n}⩾{\mathrm{N}}^{\prime }\mid {\mathrm{V}}_{\mathrm{n}}-\mathrm{b}\mid <0.....\left(1\right)$$$$\mathrm{since}\mathrm{we}\mathrm{are}\mathrm{in}\mathrm{finite}\mathrm{dimensilm}$$$$\mathrm{lets}\mathrm{defind}\mathrm{A}\mathrm{metric}?\mathrm{over}{\mathbb{R}}^2$$$$\mathrm{d}((\mathrm{x},\mathrm{y});({\mathrm{x}}^{\prime },{\mathrm{y}}^{\prime }))=\max (\mid \mathrm{x}-{\mathrm{x}}^{\prime }\mid ;\mid \mathrm{y}-{\mathrm{y}}^{\prime }\mid )$$$$\mathrm{let}\mathrm{r}>0{\beta }_{\mathrm{r}}\left(\mathrm{x}\right)\cap \mathrm{Cl}\left({\mathrm{Q}}^2\right)$$$$\mathrm{r}>0\mathrm{withe}1\mathrm{\exists }{\mathrm{N}}_1,{\mathrm{N}}_2$$$$\mid {\mathrm{u}}_{\mathrm{n}}-\mathrm{a}\mid <\mathrm{r};\mathrm{\forall }\mathrm{n}⩾{\mathrm{N}}_1$$$$\mid {\mathrm{u}}_{\mathrm{n}}-\mathrm{b}\mid <\mathrm{r};\mathrm{\forall }\mathrm{n}⩾{\mathrm{N}}_2$$$$\mathrm{tack}\mathrm{N}=\max ({\mathrm{N}}_1,{\mathrm{N}}_2);\mathrm{\forall }\mathrm{n}⩾\mathrm{N}$$$$\mathrm{d}(({\mathrm{u}}_{\mathrm{n}},{\mathrm{v}}_{\mathrm{n}});(\mathrm{a},\mathrm{b}))=\max (\mid {\mathrm{u}}_{\mathrm{n}}-\mathrm{a}\mid ;\mid {\mathrm{v}}_{\mathrm{n}}-\mathrm{b}\mid )<\mathrm{r}$$$$\Rightarrow (\mathrm{a},\mathrm{b})\in \mathrm{cl}\left({\mathrm{Q}}^2\right)$$$$\Rightarrow {\mathrm{IR}}^2\subset \mathrm{Cl}\left({\mathrm{Q}}^2\right)\Rightarrow \mathrm{equalitt}$$$$$$$$$$$$$$

Commented by mnjuly1970 last updated on 25/Feb/24

$$thanksalotsirwhicher$$

Commented by witcher3 last updated on 25/Feb/24

$$\mathrm{withe}\mathrm{pleasur}\mathrm{barak}\mathrm{alah}\mathrm{Fikoum}$$

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