Further Analysis | Part IB, 2002

(a) Given a topology T\mathcal{T} on XX, a collection BT\mathcal{B} \subseteq \mathcal{T} is called a basis for T\mathcal{T} if every non-empty set in T\mathcal{T} is a union of sets in B\mathcal{B}. Prove that a collection B\mathcal{B} is a basis for some topology if it satisfies:

(i) the union of all sets in B\mathcal{B} is XX;

(ii) if xB1B2x \in B_{1} \cap B_{2} for two sets B1B_{1} and B2B_{2} in B\mathcal{B}, then there is a set BBB \in \mathcal{B} with xBB1B2x \in B \subset B_{1} \cap B_{2}.

(b) On R2=R×R\mathbb{R}^{2}=\mathbb{R} \times \mathbb{R} consider the dictionary order given by

(a1,b1)<(a2,b2)\left(a_{1}, b_{1}\right)<\left(a_{2}, b_{2}\right)

if a1<a2a_{1}<a_{2} or if a1=a2a_{1}=a_{2} and b1<b2b_{1}<b_{2}. Given points x\mathbf{x} and y\mathbf{y} in R2\mathbb{R}^{2} let

x,y={zR2:x<z<y}\langle\mathbf{x}, \mathbf{y}\rangle=\left\{\mathbf{z} \in \mathbb{R}^{2}: \mathbf{x}<\mathbf{z}<\mathbf{y}\right\}

Show that the sets x,y\langle\mathbf{x}, \mathbf{y}\rangle for x\mathbf{x} and y\mathbf{y} in R2\mathbb{R}^{2} form a basis of a topology.

(c) Show that this topology on R2\mathbb{R}^{2} does not have a countable basis.

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