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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 2

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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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