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数学网课代修|拓扑学网课代修|topology midterm 2021

什么是拓扑学?拓扑学的历史。

拓扑一开始是一门具象学科,topology这个名字最早可能起源于Poincare的论文Papers on Topology,但是相关问题的研究,从Riemann开创Riemann Geometry和Aleksandr Lyapunov基于一些物理问题开始研究动力系统的时候就开始了。研究19世纪上半页,很多真正一流的数学家都在研究代数拓扑,如Shiing-Shen Chern,Serre, Weil, Grothendieck, Milnor,并且在20世纪上半页,这一门学科极大的被严格化,内涵被相当程度的丰富了,后续的数学发展表明,这门学科和很多看似不相关的学科有着很多不可思议的联系,而很多困难的问题最后被发现本质上是一个拓扑问题,比如有限域上Riemann猜想的解决,又或者sandwich theorem用来得到Kakeya猜想的进展都是很好的例子,另一方面,很多源于topology本身的猜想,比如Poincare猜想,被发现在原有领域是相当困难的,而被用geometric flow的方法解决,这些例子均表明topology无疑处在数学最深刻最内核的概念和对象附近。

拓扑学和其他数学课程的联系

而拓扑学同时也是一门内容丰富的学科,作为数学系本科生的基础课,至少需要掌握的应该是点集拓扑和代数拓扑,这是学习K3 Surface, Riemann Surface, 3D Topology, Algebraic Geometry等课程的的坚实的基础。

拓扑学的内容

Topological SpacesBorel Sets
Continuity, Compactness and ConnectednessMetric Spaces
Fundamental GroupHomology Group
Universal CoverCellular Homology
Universal Coefficient Theorem for CohomologyDefinition of Higher Homotopy Groups
Computation of Homology of Projective SpacesKunneth Theorem
Poincare DualitySurfaces
Complexes and Cellular HomologyHomotopy Exact Sequence of a Pair
Urysohn’s LemmaHausdorff Spaces
Quotient Topology Axiomatic Properties
Tietze Extension Theorem and ApplicationsHurewicz Theorem
Singular CohomologyRuled Surface and Conical Surface
Axiomatic PropertiesBetti Numbers and Euler Characteristics
Stoke’s TheoremWhitehead Theorem
Hurewicz HomomorphismCovering Spaces
FibrationSmooth Manifolds
Product TopologyTychonoff’s Theorem
Connected PathHomeomorphisms
Lindelof and Compact SpacesPasting Lemma
Continuous MapsUrysohn Embedding Lemma and Metrization Theorem
Baire Category TheoremSubspace Topology
Universal Coefficient Theorem for HomologySimple Computation of Homolgy Groups
Cellular HomologyProperly Discontinuous Action
Lifting PropertiesDeck Transformations
Classification TheoremPrincipal Bundles and Fibre Bundles
Borsuk-Ulam TheoremLinking Number and Index Of Vector Fields
Clutching ConstructionThe Poincare-Hopf Theorem
Mayer-Vietoris SequenceDevelopable Surface
Hausdorff TopologyDifferential Forms on Manifolds
Computation of CohomologySimplicial Complex and Simplicial Homology
Topological ManifoldsVan Kampen’s Theorem

以下是一次MATM042/3/SEMR2拓扑学考试的案例

以下是一次MATM042/3/SEMR2拓扑学考试的案例

问题 1. Consider the following cases of a set $X$ and a collection $\tau \subseteq P(X)$ of subsets of $X$ :
(a) $\quad X=\{1,2,3,4\}, \quad \tau=\{\emptyset, X,\{1\},\{1,3\},\{2,3,4\}\}$
(b) $\quad X=\{1,2,3,4,5,6\}, \quad \tau=\{\emptyset, X,\{3,4\},\{1,2,3,4\},\{3,4,5,6\}\}$
(c) $\quad X=\mathbb{R}, \quad \tau=\{\mathbb{R}, A \subseteq \mathbb{R} \mid \mathbb{R} \backslash A$ an uncountable set $\}$
In each of these cases, determine which of the three properties (O1), (O2) and (O3) of a topology are satisfied by $\tau$. Justify your answers. Hence determine whether or not $\tau$ defines a topology on the given set $X$. [12 marks]
证明 . 按照拓扑的定义直接验证
问题 2.

(a) Let $(X, \tau)$ be a topological space. Let $A \subseteq X .$ Let $x \in A$.
Let $\mathfrak{B}(x)$ be a neighbourhood basis of $x$ in the topology $\tau$ on $X$.
Prove that
$$
\{B \cap A \mid B \in \mathfrak{B}(x)\}
$$
is a neighbourhood basis of $x$ in the induced topology $\left.\tau\right|_{A}$ on $A$. marks]
(b) Let $X=\mathbb{R}$. For each $x \in \mathbb{R}$, let
$$
\mathfrak{B}(x):=\{(y, x] \mid y \in \mathbb{R}, y<x\}
$$
be a neighbourhood basis of $x$ which defines the topology $\tau$ on $\mathbb{R}$. Let $A=[0,1] \subseteq \mathbb{R}$. Use the result in (a) to find a neighbourhood basis of a point $x \in A$ in the induced topology $\left.\tau\right|_{A}$ on $A$.
$\{$ Hint: Consider the cases $x=0$ and $0<x \leq 1$ separately, and simplify your results. $\}$ $[5$ marks $]$

证明 . 按照子空间拓扑的定义和拓扑邻域基的概念验证
问题 3.

Consider the topological space $(\mathbb{R}, \sigma)$ with standard metric topology $\sigma$ on $\mathbb{R}$. Prove that the topological space $(\mathbb{R}, \sigma)$ is second countable. $\quad$ [6 marks]

证明 . 容易证明有理数为中心有理数为半径的集合是一组拓扑基
问题 4.

Let $(X, \tau)$ be a topological space. Let $A \subseteq X$ and $B \subseteq X$. Use the definition of the interior of a subset to prove that
$$
(A \cup B)^{\circ} \supseteq A^{\circ} \cup B^{\circ}
$$
Determine whether or not
$$
(A \cup B)^{\circ} \subseteq A^{\circ} \cup B^{\circ}
$$
If so, give a proof. If not, give a counterexample.

证明 .

拓扑可以很稀疏所以反例是显然存在的

问题 5.

Let $X=\mathbb{R}^{2}$ with standard metric topology $\tau$ on $\mathbb{R}^{2},$ and let $Y=\mathbb{R}$ with standard metric topology $\sigma$ on $\mathbb{R}$. Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be the map
$$
f(x, y):=x+y \quad \text { for } \quad(x, y) \in \mathbb{R}^{2}
$$
Prove that the function $f$ is continuous. Determine whether or not $f$ is a homeomorphism. Justify your answer. $\quad$ [12 marks]

证明 .

按照定义直接验证

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