# Prove there are open sets in the lower limit topology, that are not open in with the absolute value metric

What would be an example of an open set in the lower-limit topology that isn't open with the absolute value metric over the real numbers.

Further, how would I show that the lower limit topology is not a discrete topology?

I'm assuming that for the discrete topology, I could show that there is no singleton set in the lower-limit topology, hence it can't be discrete.

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## 2 Answers

Sets of the form $[a,x)$ are open in the lower-limit topology by definition (at least in mine). You can show that they are not open in the metric topology as there is no neighborhood about $a$ such that $(a-\epsilon,a+\epsilon)\subseteq [a,x)$ for any $\epsilon>0$.

To show that the topology is not discrete, you can show that singletons are not open. And this is true because there is no $\epsilon>0$ such that $[a,a+\epsilon)\subseteq\{a\}$.

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Hint: What are the basic open sets of the lower limit topology? Can you find a point of such a set that is not interior to this basic set in the absolute value-induced topology?

You do indeed want to show that no singleton is open in the lower limit topology.

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