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Question

Above is my question. My only issue is the final bit! For statements $1.$ and $2.$, the answer is true, since in both cases $Y$ is normal and we know that both metric and compact, Hausdorff spaces are normal. I'm assuming that $3.$ is false: firstly, it looks false (require normality for "standard" Urysohn, so unlikely to work for arbitrary topological space in the new case) and also the other two are true (unlikely to be three true!). However, I can't for the life of me come up with a counter example!

We know that for Urysohn if we define $$U = \{ x \in K \vert f(x) < 1/2 \} \, \ V = \{ x \in K \vert f(x) > 1/2 \},$$ then we obtain open, disjoint sets with $U \supseteq E$ and $V \supseteq F$ (where $E$ and $F$ are the closed, disjoint sets in Urysohn). This is for any such $E$ and $F$. So I thought that if I just define $X$ to be a non-normal space (eg, $X = \{ 0, 1, 2, ... \} = \Bbb N$), then I'd just be done since Urysohn fails... but in every case that I've tried while Urysohn fails, the extension property holds!

Any help would be most appreciated! As always, hints please, not just solutions!

Sam OT
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    Robert Israel’s hint is good. You could also start with any non-normal space and let your closed set by $A\cup B$, where $A$ and $B$ are disjoint closed sets that cannot be separated by disjoint open sets. Two explicit examples, one of them Tikhonov, are given in the answers to this question. And at some point you might be interested in this answer, which contains a description of a $T_3$ space that has two points that cannot be separated by ... – Brian M. Scott Apr 20 '15 at 18:27
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    ... a continuous real-valued function, and the answers to this question, which give another such space and a way to use such spaces to construct $T_3$ spaces on which all continuous real-valued functions are constant. – Brian M. Scott Apr 20 '15 at 18:29
  • Ah yes, the three point space, as shown in the first link. I was rather struggling to come up with a non-normal topological space. I don't know anything about the $T_1$ or $T_3$ (Tikhonov?) spaces (not even the definitions). The course is a course in Linear Analysis, not topology. Thanks for your help. – Sam OT Apr 20 '15 at 22:10
  • You’re welcome. – Brian M. Scott Apr 20 '15 at 22:12

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Hint: take your favourite non-Hausdorff topological space $X$, and let $S$ consist of two points that do not have disjoint neighbourhoods.

Robert Israel
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