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And we continue our parade of counterexamples! Most of them are again in the field of real analysis, but I put some other stuff in there as well.

This time the format is a bit different. We present 10 statements that are all of the nature ##P## if and only if ##Q##. As it turns out, only one of those implications is really true, the other is not. The objective is to both prove the true implication and provide a counterexample to the false one. There are two catches however:

This time the format is a bit different. We present 10 statements that are all of the nature ##P## if and only if ##Q##. As it turns out, only one of those implications is really true, the other is not. The objective is to both prove the true implication and provide a counterexample to the false one. There are two catches however:

- There is one statement where both implications are completely true. In this case, you must prove both statements.
- There is one statement where both implications are completely false. In this case, you must provide a counterexample to both statements.

- For an answer to count, the answer must not only be correct, but a detailed argumentation must also be given as to why it is correct.
- Any use of outside sources is allowed, but do not look up the question directly. For example, it is ok to go check analysis books, but it is not allowed to google the exact question.
- If you previously encountered this statement and remember the solution, then you cannot participate in this particular statement.
- All mathematical methods are allowed.
- The first person to provide a complete answer will be credited. Other people may be credited for the answer as well, depending on the contribution they have given.

- SOLVED BY mfb Given a series ##\sum a_n##. Then the ratio test can be used to establish convergence of the series if and only if the root test can be used to establish convergence of the series.

- SOLVED BY andrewkirk In any Banach space ##X## and given any series ##\sum a_n## in ##X##, then the series converges absolutely (that is: ##\sum \|a_n\|## converges) if and only if the series converges unconditionally (that is: for any bijection ##\pi:\mathbb{N}\rightarrow \mathbb{N}## holds that ##\sum_n a_{\pi(n)}## converges to the same number).

- SOLVED BY Samy_A A set ##A\subseteq \mathbb{R}^2## is closed if and only if it is the topological boundary of some set. That is: there is some ##B\subseteq \mathbb{R}^2## such that ##\partial B = A##. https://en.wikipedia.org/wiki/Boundary_(topology)

- SOLVED BY fresh_42 For a number ##n\in \mathbb{N}\setminus \{0,1\}## holds that there exists (up to isomorphism) only one group of order ##n## if and only if ##n## is prime.

- SOLVED BY mfb A function ##f:\mathbb{R}\rightarrow \mathbb{R}## is continuous if and only if it is almost everywhere equal to a continuous function.

- SOLVED BY Samy_A A function ##f:\mathbb{R}\rightarrow \mathbb{R}## is constant if and only if it is differentiable almost everywhere and ##f^\prime = 0## almost everywhere.

- SOLVED BY Samy_A A function ##f:\mathbb{R}\rightarrow \mathbb{R}## is Borel measurable if and only if it is the pointwise limit of continuous functions.

- SOLVED BY Samy_A A compact topological space ##X## is separable if and only if each collection of pairswise disjoint open sets is countable.

- SOLVED BY mfb A function ##f:\mathbb{R}\rightarrow \mathbb{R}## is measurable if and only if it is somewhere differentiable of order ##2##.

- SOLVED BY fresh_42 A field ##F## is infinite if and only if it has zero characteristic.

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