# Linked Questions

33answers
14k views

### 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?
32answers
44k views

### 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 ...
7answers
4k views

### 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 ...
2answers
15k views

### Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago when I studied in the university I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows: All numbers are divided into two classes: those ...
8answers
8k views

### 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 $R=262537412640768744 - \epsilon$ ...
2answers
4k views

### 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 ...
5answers
3k views

I am asking for a way to compute the rank of the 'join' of a bunch of subspaces whose pairwise intersections might be non-zero. So in the case n=2 this is just $\dim(A_1+A_2) = \dim(A_1) + \dim(A_2) - ... 3answers 6k views ### When is$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)$is separable? In reality, I am interested in ... 5answers 3k views ### Projection of Borel set from$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 7answers 1k views ### Extensional theorems mostly used intensionally 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 3k 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 ... 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 ... 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 3k 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 ...

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