Prove: For any topologies T_{1} and T_{2} for X, if B_{1} is a base for T_{1}, and if B_{1} ⊂ T_{2}, then T_{1} ⊂ T_{2}.
Using:
Thm 2.1: A is open in (X,T) iff For every p ∈ A, there is G ∈T such that p ∈ G and G ⊂ A
Cor 8.2: B is a base for T iff: B ⊂ T and for every G ∈ T, if p ∈G, then there is A ∈ B such that p is an element of A ⊂ G
can somebody help me with these two problems immediately???
1.Let D_p be any open disc in R×R with centre p=(a,b) and radius δ>0.Show that there exist an open disc (Dbar) such that
(i)the centre of (Dbar) has rational coordinates.
(ii) the radius of (Dbar0 is rational.
(iii)p is an element of (Dbar) and (Dbar) is a subdet of D_p
2.Let f:R→R be a continuous function such that f(q)=sinq for q∈Q.Find the value of f(π/4).