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Consider the set R1of all real-valued functions on the unit interval.
Consider the set R1of all real-valued functions on the unit interval.1. For each f Є R1, each finite subset F of I and each positive δ , letU(f, F, δ ) = {g Є R1 |g(x) - f(x)| < δ, for each X Є F}.Show that the sets U(f, F, δ ) form a neighborhood base at f, making R1 a topological space.2. For each f Є R1 the closure of the one-point set { f } is just { f }. (This is not unusual. In fact, it is a situation to be desired; spaces without this property are difficult to deal with)3. For f Є R1 and ε > 0, letV(f, ε) = {g Є R1 lg(x) - f(x)l < ε, for each x Є I }.Verify that the sets V(f, ε) form a neighborhood base at f, making R1 a topological space.4. Compare the topologies defined in 1 and 3.
