Open Set And Closed Set

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  • #138

    AdminCSEC
    Keymaster

    Let G be an open subset of ℝ.
    If 0 ∈ G and if x + y ∈ G for all x, y ∈ G, then show that G = ℝ.

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    #275

    ramKS
    Participant

    Sol. Since 0 ∈ G and G is open, there exists ε > 0 such that
    B(0, ε) ⊆ G (where B(0, ε) := {x ∈ ℝ | |x| < ε}).

    Now let x ∈ ℝ arbritary. There exists k ∈ N such that ∣x/k∣ < ε, hence x/k ∈ B(0, ε) ⊆ G. Now we can write

    x = x/k+�+x/k (k times)

    and conclude x ∈ G. (Since, G is closed under addition).
    Hence ℝ ⊆ G.

    G ⊆ ℝ (given).
    thus, G = ℝ.

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