1. ## Topology Proof

Prove: For any topologies T1 and T2 for X, if B1 is a base for T1, and if B1 ⊂ T2, then T1 ⊂ T2.

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

2. ## Re: Topology Proof

Originally Posted by Kiefer
Prove: For any topologies T1 and T2 for X, if B1 is a base for T1, and if B1 is a subset of T1, then T1 is a subset of T2.
Please review your posting. I think that you have a misplaced subscript.
Do you mean $B_1\subset T_2~?$

3. ## Re: Topology Proof

Originally Posted by Kiefer
Prove: For any topologies T1 and T2 for X, if B1 is a base for T1, and if B1 ⊂ T2, then T1 ⊂ T2.
Thank you for the edit.
If $O\in T_1$ then then is a collection of sets $B_{\alpha}$ from $B$ such that $O = \bigcup\limits_\alpha {B_\alpha }$ (basis)

Because $T_2$ is a topology and $B\subset T_2$ we know that $O = \bigcup\limits_\alpha {B_\alpha } \in T_2$.

Proof done.

4. ## Re: Topology Proof

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)=sin⁡q for q∈Q.Find the value of f(π/4).