List questions from examples-counterexamples

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If a triangle is not equilateral, must its orthocenter and circumcenter be distinct?

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quotient topology doesn't preserve separation axioms

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tangent plane to level set

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$A$ so that $\lim\limits_{n \to \infty}\frac{1}{n} \xi_{\big| \Bbb{N}} (A \cap[1,n])$ is not defined. ($\xi$ = counting measure)

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A topological property that holds by countable union/intersection but not by uncountable union/intersection?

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Does there exist a function such that the following criteria are satisfied simultaneously? Can you provide an example?

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Finding a counterexample for a statement about integrals of function series

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Is there an ideal of $(C^0(\mathbb{R}),+,\cdot)$ that isn't prime?

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Is there a maximal ideal that isn't prime if the ring is non commutative?

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Finding a bounded function with unbounded derivative at zero

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Is a surjective continuous map with compact domain is open?

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List Question

Example of such a module

$Proj S$ integral scheme with S not a domain?

If a triangle is not equilateral, must its orthocenter and circumcenter be distinct?

quotient topology doesn't preserve separation axioms

tangent plane to level set

Primality criterion

An example of a direct product that is not a Cartesian product

$A$ so that $\lim\limits_{n \to \infty}\frac{1}{n} \xi_{\big| \Bbb{N}} (A \cap[1,n])$ is not defined. ($\xi$ = counting measure)

A topological property that holds by countable union/intersection but not by uncountable union/intersection?

Does there exist a function such that the following criteria are satisfied simultaneously? Can you provide an example?

Finding a counterexample for a statement about integrals of function series

Is there an ideal of $(C^0(\mathbb{R}),+,\cdot)$ that isn't prime?

Is there a maximal ideal that isn't prime if the ring is non commutative?

Finding a bounded function with unbounded derivative at zero

Is a surjective continuous map with compact domain is open?

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