Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $\mathbb{R}_l$ be the real line with the left closed interval topology (= the topology on $\mathbb{R}$ with base consisting of all intervals $[a, b)$ with $a < b$). We will consider $\mathbb{R}_l \times \mathbb{R}_l$ with the product topology (= Sorgenfrey plane). This problem shows that this space is not normal. The argument is by contradiction. So, assume that the Sorgenfrey plane is normal.

Let $L \subset \mathbb{R}_l \times \mathbb{R}_l$ be given by $L = {(t, -t) | t \in \mathbb{R}}$

Let $D = \mathbb{Q} \times \mathbb{Q} \subset \mathbb{R}_l \times \mathbb{R}_l$ be the set of all points with both coordinates rational.

a. Show: if $A \subset L$, then A is a closed subset of the Sorgenfrey plane.

Now assume that for each $A \subset L$ we can choose (and fix) open subsets $U_A, V_A$ in $\mathbb{R}_l \times \mathbb{R}_l$ such that $A \subset U_A, L - A \subset V_A$, and $U_A \cap V_A = \varnothing$. Define a function $$ \vartheta : P(L) \to P(D) $$

by setting $\vartheta(A) = U_A \cap D$.

b. Show that $\vartheta(A)$ determines $A$. Hint: think of sequences in NE quadrant.

c. Why does this show that $\vartheta$ is an injection?

d. Why does this contradict a result of Cantor?

share|improve this question
    
What have you tried? –  Thomas E. Feb 27 '13 at 5:33

1 Answer 1

HINTS:

(a) Just show that each point of $\Bbb R_\ell^2\setminus L$ has an open nbhd disjoint from $L$; this is completely trivial.

(b) What you’re being asked to show is that if $\theta(A)=\theta(B)$, then $A=B$ or, equivalently, that if $A,B\subseteq L$ and $A\ne B$, then $\theta(A)\ne\theta(B)$. Suppose that $a\in A\setminus B$. Then $a\in U_A\cap V_B$, and therefore $\theta(A)\cap V_B\ne\varnothing$; why? And this in turn implies (why?) that $\theta(A)\setminus\theta(B)\ne\varnothing$ and hence that $\theta(A)\ne\theta(B)$.

(c) Trivial.

(d) How many subsets does $D$ have? How many subsets does $L$ have?

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.