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prove-that-X-R-e-is-a-second-topology-space-e-is-Euclidian-topology-on-R-Hint-B-r-1-n-r-1-n-r-Q-n-N-is-a-base-for-e-




Question Number 140407 by mnjuly1970 last updated on 07/May/21
          prove that ⟨ X:=R , τ_e  ⟩ is        a second topology space .       τ_e   is Euclidian topology on R.     Hint ::  B= { (r−(1/n) ,r+(1/n))∣ r∈Q , n∈N}     is  a base for τ_(e ) .....
$$\:\:\:\: \\ $$$$\:\:\:\:{prove}\:{that}\:\langle\:\mathrm{X}:=\mathbb{R}\:,\:\tau_{{e}} \:\rangle\:{is} \\ $$$$\:\:\:\:\:\:{a}\:{second}\:{topology}\:{space}\:. \\ $$$$\:\:\:\:\:\tau_{{e}} \:\:{is}\:\mathscr{E}{uclidian}\:{topology}\:{on}\:\mathbb{R}. \\ $$$$\:\:\:\mathrm{H}{int}\:::\:\:\mathcal{B}=\:\left\{\:\left({r}−\frac{\mathrm{1}}{{n}}\:,{r}+\frac{\mathrm{1}}{{n}}\right)\mid\:{r}\in\mathrm{Q}\:,\:{n}\in\mathbb{N}\right\} \\ $$$$\:\:\:{is}\:\:{a}\:{base}\:{for}\:\tau_{{e}\:} ….. \\ $$

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