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### Most harmful heuristic?

What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?
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### Why do we teach calculus students the derivative as a limit?

I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students? Something a teacher ...
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### When does Cantor-Bernstein hold?

The Cantor-Bernstein theorem in the category of sets (A injects in B, B injects in A => A, B equivalent) holds in other categories such as vector spaces, compact metric spaces, Noetherian topological ...
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### Why are powers of $\exp(\pi\sqrt{163})$ almost integers?

I've been prodded to ask a question expanding this one on Ramanujan's constant $R=\exp(\pi\sqrt{163})$. Recall that $R$ is very close to an integer; specifically $R=262537412640768744 - \epsilon$ ...
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### Cohomology and fundamental classes

Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...
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### When is $L^2(X)$ separable?

I have never studied any measure theory, so apologise in advance, if my question is easy: Let $X$ be a measure space. How can I decide whether $L^2(X)$ is separable? In reality, I am interested in ...
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### Projection of Borel set from $R^2$ to $R^1$

Hello This should be easy to prove but i have no idea how to do it: If $X \subseteq \mathbb{R}^2$ is borel then $f(X)$ is borel where $f(x,y) = x$ Thanks Tobias
Some theorems are stated and proved extensionally, but in practice are almost always used intensionally. Let me give an example to make this clear -- integration by parts: $$\int_a^b f(x)g'(x)ds = ... 4answers 2k views ### Cocomplete but not complete abelian category This is a duplicate of the following question to which I did not receive any answer: http://math.stackexchange.com/questions/238247/complete-but-not-cocomplete-category Let \mathfrak C be an ... 3answers 3k views ### Is “compact implies sequentially compact” consistent with ZF? Over at the nForum, we've been discussing sequential compactness. The discussion led me to realise that I naively assumed that nets were simply Big Sequences, and that I could make a reasonable guess ... 2answers 1k views ### When is sin(r \pi) expressible in radicals for r rational? Perhaps this question will not be considered appropriate for MO - so be it. But hear me out before you dismiss it as completely elementary. As the question suggests, I would like to know when ... 2answers 2k views ### most general way to generate pairwise independent random variables? Is there a sort of structure theorem for pairwise independent random variables or a very general way to create them? I'm wondering because I find it difficult to come up with a lot of examples of ... 1answer 1k views ### Set Theory and V=L From http://en.wikipedia.org/wiki/Analytical_hierarchy "If the axiom of constructibility holds then there is a subset of the product of the Baire space with itself which is \Delta^1_2 and is the ... 3answers 302 views ### Non-continuous higher differentiability The standard definition is that a function f:\mathbb{R}^n\to \mathbb{R} is differentiable at a point x if there exists a linear map \mathrm{d}f_x: \mathbb{R}^n \to \mathbb{R} such that$$f(x+h) ...
Hi, Could anyone give an example such that: $$Y_i \rightarrow Y_{\infty}, \text{a.s.},$$ and $Y_i$'s are uniformly integrable. But $\mathbb{E}(Y_i|\mathcal{G})$ does not converge a.s. to ...