funny problem from model theory problem sheet 题面 Let L be a countable language, T a theory. Let c_1,c_2,... be new constants, L'=L\cup\lbrace c_1,...\rbrace, and let X be the set of all complete theories of L' with T\subset t. Enumerate the L' se……

Rotman algebraic topology上的题, 个人认为出的非常好, 极大增进了读者对\pi_1(S^1)计算过程的理解. 题面 A group G which is also a topology space is called a topological group if \mu:G\times G\to G:(x,y)\mapsto xy and i:G\to G:x\mapsto x^{-1} are contin……

Dieudonne的题, 真实难度不好说, 回顾一下发现其实挺简单的. 题面 Let E be a Hilbert space, F a dense linear subspace of E, distinct from E. (For example E space of l^2 sum of a suitable countable linear independent set and F space of finite sum of thi……

题目背景 证明连续性时缩放水平哈哈了(sweatgrinning) 题面 f is a nonconstant holomorphic function on a region \Omega_1 and g is defined on f(\Omega_1). Prove that if h=g\circ f is holomorphic, then so is g. 先想再看提示哦 提……

题目背景 完整题解来自ox帅气lecturer, 我只能证明F<\mathbb C的情况 ;( ;( ;() 题面 V是域F上的一个线性空间, \dim V=n, 且有T\in L(V). 现取\lambda_1,...,\lambda_n\in F两两不等, 且任意\lambda_i都不是T的特征值. 试证明存在\alpha_1,...,\alpha_n\in F使得 \……

题目背景 人见人爱的代数小题 :] 惊人的是竟然在Hungerford II.2.8.9才出现, 还标了名字 😮 题面 A^ * is a normal subgroup of A, B^ * is a normal subgroup of B. Prove that: (i) A^ *(A\cap B^ *) is a normal subgroup of A^ *(A\cap B) (ii) B^ *(A^ *\cap B) i……

题目背景 本题竟然与xgn目前正在学习的序数有关, 真是可喜可贺!(crim boi在thu里有包含序数的课程吗?) 题面 If you don't remember what a topology is, please refer to this passage. Preliminary: Open sets in \mathbb R are just union (can be infinite) of som……

题目背景 Rudin Real and Complex Analysis第一题, 真是开门红. 题面 We call a pairing (X,\mathfrak{M}) \sigma-algebra if \mathfrak{M} is a subset of P(X) satisfying the three properties below: (1) X\in\mathfrak{M} (2) If A\in\mathfrak{M}, A^c\in\mathf……

题目背景 南大几把选拔中最水的题(T3), 适合出在初中竞赛里. 另外这期写解答方便以后想考南大几把的学生?, 篇幅太长除了题面不写中文了. 不过题面也是pmt提供的, 不一定完全准确. 题面 记排列P的置换环数量为c(P),排列P,Q的复合记为P\oplus Q，已知长度为n的排列f_1,f……

题目背景 贴吧范畴论帅哥问的一道题, 题目还算基础, 不过确实很容易伪证, 是名副其实的脑筋急转弯. 题面 对一个拓扑空间X, 定义集合S为稠密集当且仅当\text{Cl}(S)=X, R为无处稠密集当且仅当X-\text{Cl}(R)为稠密集. 定义集合A的边界\text{Bd}(A)=\text{Cl}(A)\cap\te……