\(\underline{Def:}\)A topology space \(\mathcal{X}=(\underline{X},\eth_{x})\)consists of a set \(\underline{X}\),called the underlying space of \(\mathcal{X}\) ,and a family \(\eth_{x}\)of subsets of \(\mathcal{X}\)(ie.\(\eth_{x}\subset P(\underline{…

In James Munkres "Topology" Section 22, the quotient space is defined as below. Definition Let \(X\) be a topological space, and let \(X^*\) be a partition of \(X\) into disjoint subsets whose union is \(X\). Let \(p: X \rightarrow X^*\) be the…

目录 引入:绝对值 度量空间 Example: 开集,闭集 引入:绝对值 distance\(:|a-b|\) properties\(:(1)|x| \geq 0\),for all \(x \in R\),and \("=" \Leftrightarrow x=0\) \((2):|a-b|=|b-a|(|x|=|-x|)\) \((3):|x+y| \leq |x|+|y|\),for all \(x,y \in R\) (\(|a-c| \leq |a-b|+|b-c|\))…

http://www.fxguide.com/featured/pixars-opensubdiv-v2-a-detailed-look/ Pixar’s OpenSubdiv V2: A detailed look By Mike Seymour September 18, 2013 Subdivision is key modeling tool that allows greater accuracy and the OpenSubdiv project aims to standardi…

What: 就是将统计学算法作为理论,计算机作为工具,解决问题.statistic Algorithm. How: 如何成为菜鸟一枚? http://www.quora.com/How-can-a-beginner-train-for-machine-learning-contests 链接内容总结: "学习任何一门学科,framework是必不可少的东西.没有framework的东西,那是研究." -- Jason Hawk One thing is for sure; you ca…

Arithmetic operations taught in elementary schools are continuous in the high level topological point of view. This signifies that there is literally no clear boundary between simple and complex, low and high concepts. Instead, they both play indispe…

Lemma 21.2 (The sequence lemma) Let \(X\) be a topological space; let \(A \subset X\). If there is a sequence of points of \(A\) converging to \(x\), then \(x \in \bar{A}\); the converse holds if \(X\) is metrizable. Proof a) Sequence convergence \(\…

Proof of Theorem 20.3 Theorem 20.3 The topologies on \(\mathbb{R}^n\) induced by the euclidean metric \(d\) and the square metric \(\rho\) are the same as the product topology on \(\mathbb{R}^n\). Proof: a) Prove the two metrics can mutually limit ea…

Understanding of continuity definition in topology When we learn calculus in university as freshmen, we are usually force-fed with the \(\epsilon-\delta\) language for the definition of a function’s continuity, i.e. A function \(f: A \rightarrow \mat…

(Remark: The proof presented in this post is a reorganization and interpretation of that given by James Munkres in his book "Topology".) Theorem 37.3 (Tychonov Theorem) The product of arbitrary number of compact spaces is compact in the product…