# Ben

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From the Bay to LA.

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 Dec8 awarded Caucus Dec8 revised Without AC, it is consistent that there is a function with domain $\mathbb{R}$ whose range has cardinality strictly larger than that of $\mathbb{R}$? added 44 characters in body Dec7 awarded Scholar Dec7 accepted Without AC, it is consistent that there is a function with domain $\mathbb{R}$ whose range has cardinality strictly larger than that of $\mathbb{R}$? Dec7 comment Without AC, it is consistent that there is a function with domain $\mathbb{R}$ whose range has cardinality strictly larger than that of $\mathbb{R}$? Pretty interesting, thanks Asaf. Dec7 comment Without AC, it is consistent that there is a function with domain $\mathbb{R}$ whose range has cardinality strictly larger than that of $\mathbb{R}$? @BrianM.Scott Oops, thanks for clarifying that. Dec7 awarded Student Dec7 asked Without AC, it is consistent that there is a function with domain $\mathbb{R}$ whose range has cardinality strictly larger than that of $\mathbb{R}$? Dec7 answered Number of non-isomorphic groups of order $p^2$ Dec7 comment Number of non-isomorphic groups of order $p^2$ It's a bit of a sledgehammer here, but do you know the structure theorem for finitely generated abelian groups? Dec4 comment Prove that a $C^\infty$ vector field on $M$ can be extended to a $C^\infty$ vector field on $N$ . When you say you have a smooth vector field $X$ on $M$, are you assuming that for each $p\in M$, there exists an open neighborhood $V$ of $p$ in $N$ and a vector field $\tilde{X}$ on $V$ which agrees with $X$ on $V\cap M$? Nov29 answered Conjugacy Class of a Group of Order 12 Nov29 answered Finite field extensions and minimal polynomial Nov24 answered Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups Nov24 revised $A$ and $A+y$ are homeomorphic where $A$ is open set edited tags Nov23 comment Circle to circle homotopic to the constant map? Are you asking for the existence of some $x$ such that $f(x)=-x$ or that $f(x)=-x$ for all $x$? Nov23 comment Circle to circle homotopic to the constant map? What is $x$ in part (b)? Nov18 revised $S\subseteq V \Rightarrow \text{span}(S)\cong S^{00}$ fixed typo Nov14 revised Explicit description of a topology that proves $\mathbb{R}$ is not minimal Hausdorff. edited title Nov2 comment Homotopy equivalence for $S^n$ with finite k punctures The basic idea is to view $S^n\setminus\{k\text{ points}\}$ as a punctured sphere with $k-1$ other missing points. By stereographic projection (there are known formulas for this), this is homotopy equivalent to $\mathbb{R}^n$ minus $k-1$ points. The punctured plane is then homotopy equivalent to the wedge of $k-1$ copies of $S^{n-1}$. Geometrically, fix some base point, and then draw $k-1$ loops out from this point that encircle exactly one of the missing points. The rest of $\mathbb{R}^n$ deformation retracts onto these loops, and this is equivalent to the wedge sum.