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1 Primary & Middle School Mathematics1–9 年级在线 2 High School Mathematics10–12 年级在线 3 University Mathematics — Year 1本科一年级在线 4 University Mathematics — Year 2本科二年级在线 5 University Mathematics — Year 3本科三年级在线

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1 Markets I: The Ecosystem and Exchange-Traded Markets市场在线 2 Markets II: Rates, FX and Credit市场在线 3 Markets III: Commodities, Energy and Crypto市场在线 4 Quantitative Methods方法在线 5 Derivatives and Volatility衍生品在线 6 Rates, Credit, XVA and Risk利率、信用与风险在线 7 Research Craft: Predictors, Backtests, Measurement, Portfolios研究在线 8 Strategies I: Equities and Futures策略在线 9 Strategies II: Volatility, Relative Value, Macro and the Bank Desks策略在线 10 Microstructure and Execution交易执行在线 11 Market Making and High-Frequency Trading做市在线 12 Machine Learning for Markets机器学习在线 13 Low-Latency Software技术在线 14 Networks, Hardware and Trading Infrastructure技术在线 15 Research, Data and Risk Platforms技术在线 16 The Desk and the Firm公司运营在线 17 The Industry: Firms, Roles and Careers职业发展在线 18 Interviews I: Maths, Probability and Speed职业发展在线 19 Interviews II: Markets, Trading Games and the Final Round职业发展在线 20 Interviews III: Code职业发展在线

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Mathematics · 术语表

什么是 compact space？

Definition 6.12 University Mathematics — Year 3 · 第 6 章 — General Topology

An open cover of $X$ is a family $(U_i)_{i\in I}$ of open sets with $\bigcup U_i = X$ . $X$ is compact if it is Hausdorff and every open cover admits a finite subcover. Equivalently (taking complements): every family of closed sets with the finite intersection property (all finite subfamilies have nonempty intersection) has nonempty total intersection.

示例

Example 6.11

The quotient $\R/\Z$ (identify $x$ and $x + n$ ) is homeomorphic to the circle $S^1 = \{z \in \C : \abs z = 1\}$ : the map $x \mapsto \eu^{2\iu\pi x}$ passes to a continuous bijection $\R/\Z \to S^1$ (Proposition 6.10(b)); its inverse is continuous by the compactness argument of Corollary 6.14 below (Exercise 6.5 details everything, including why $\R/\Z$ is Hausdorff and compact). Likewise $[0,1]$ with endpoints glued is $S^1$ , the square with opposite sides glued is the torus, and gluing is finally a theorem, not a picture.

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