[Notes] General Topology

Textbook: Topology 2e, Munkres

這是我自己讀拓樸學的筆記，我自己覺得整理得可以作為參考書了 \o\。

原書中每一章節會介紹一個主題，一個定理接一個定理證，可是我筆記時綜整一個章節的性質(我認為這些要叫 propositions/properties比較恰當)寫在一起，證明後附，因為拓樸學的 prop 很多都是定義寫出來兜一兜就會對的，把注意放在證明內容其實有點失焦，重要的是要熟知什麼樣的空間有什麼性質。

承上，最後有個連結 Review of the Basis

考慮19個「拓樸空間可以有的性質」，對每個性質 回答 20 個問題，

共 380 個小題之中，只有兩題仍是 open problem 作不出來。

真正的算過一遍，就對一般拓樸自我感覺良好通透! 可以去試試看

Notes 1 Topological Spaces basis/subbasis 2 Fundamental Topologies order topology dictionary order product topology projection subspace topology convex ordered square 3 Basic Point Set Topology closed set/closure/interior/nbh/boundary/limit point Kuratowski's Closure Complement Problem Hausdorff space T1-axiom 4 Continuous Functions Equivalent definitions Constructions local formulation pasting lemma homeomorphism local finite 5 Infinite Product Topologies box/product topology 6 Metric spaces metrization theorems uniform metric/topology $l^2$ topology continuity on metric spaces convergent sequence definition sequence lemma uniform convergent uniform limit theorem 7 Quotient topology quotient map/topology/space continuous functions out of quotient space   Notes 8 connectness connectness totally disconnected $\mathbf{R}$ 上 connected spaces linear continuum Intermediate Value Theorem path connectness $n$th root function Long Line minimal uncountable well-ordered set local (path) connectness quasicomponent 9 compactness compactness finite intersection property graph of a function perfect map $\mathbf{R}^n$ 上 compact spaces Maximum/Minimum Value Theorem Lebesgue Number Lemma Uniform Continuous Theorem Cantor set limit point compactness(Bolzano-Weierstrauss property) countable compactness sequential compactness Isometry Local compactness quoteint map (revised) 10 Countability Axioms 1st countability 2nd countability Lindelof spaces separable spaces   Notes 11 Separation Axioms regular spaces (T3) normal spaces (T4) completely normal spaces (T5) perfectly normal spaces (T6) Urysohn Lemma Completely regularity Urysohn Metrization Theorem Imbedding theorem local metrizability Tietze Extension Theorem Universal Extension Property Coherent topology 12 Manifolds partition of unity finite partition of unity existence theorem imbedding theorem for manifolds Shrinking lemma 4S Review of the basics 13 Tychonoff Theorem Tychonoff Theorem compactification induced compactification Stone-Cech compactification Review of the Basics