HerrWarum
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How can I prove formally that the projective plane is a Hausdorff space?
5 votes

Recall that an equivalence relation $\sim$ on a topological space X is said to be $\textit{open}$ if for every open subset $A$ of $X$, the set $$[A] := \{x \in X | x \sim a \text{ for some a} \in A\}$$...

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$\sum_1^\infty a_n$ converges iff $\sum_0^\infty 2^ka_{2^k}$ converges (Rudin)
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2 votes

Answer to question 1: Observe that since $$a_1 \geq a_2 \geq a_3 \cdots \geq 0,$$we have $$a_3 + a_4 \geq a_4 + a_4,$$ and similarly $ \underbrace{(a_{2^{k-1}+1}+...+a_{2^k})}_{\text{$2^{k-1}$ terms}}...

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Help! I made two groups of order 5
2 votes

In the non abelian group that you wrote down, we have $$a_1 a_2 = a_3$$ and $$a_1 a_3 = a_4.$$ Using the first relation in the second gives $$a_1 a_1 a_2 = a_4$$ i.e, $$a_2 = a_4$$ which contradicts ...

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How to prove that $M_{k-12} \to S_k$ is an isomorphism?
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0 votes

For Q1, $\Delta$ is non-vanishing on the upper half plane: indeed, since $\Delta$ is a cusp form of weight $12$, using the valence formula (aka the $k/12$ formula) we get $ord_{\infty}(\Delta) = 1$, ...

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Is a continuous function integrable in a Jordan measurable subset?
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Yes. The boundary of a Jordan measurable set is a measure zero set. Consider a closed rectangle $R$ that contains the given Jordan measurable set $A$ (boundedness is assumed to be part of the ...

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