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## MATH4530 Topology课程简介

Topology is a branch of mathematics that studies the properties of spaces that are preserved under continuous transformations, such as stretching, bending, and twisting. It is often described as qualitative geometry because it focuses on the properties of spaces that are not dependent on specific measurements or coordinates.

This course starts with basic point-set topology, which is concerned with the properties of sets and the relationships between them. Some of the topics covered in this section may include connectedness, compactness, and metric spaces. Connectedness refers to the idea that a space is made up of one piece, while compactness relates to the idea that a space has no “holes” or “gaps”. Metric spaces involve using a distance function to measure the distance between points in a space.

The later topics of the course may include the classification of surfaces such as the Klein bottle and Möbius band. These are examples of non-orientable surfaces, which means they do not have a consistent “up” and “down” orientation. Elementary knot theory, which studies the properties of knots and their classifications, may also be covered. The fundamental group and covering spaces are important concepts in algebraic topology and relate to the ways in which spaces can be mapped onto each other.

Overall, this course will introduce students to the foundational concepts of topology and its applications in various fields of mathematics.

## PREREQUISITES

The course prerequisites indicate that students must have completed one of the following courses: MATH 2210, MATH 2230, MATH 2310, or MATH 2940. Additionally, students must have taken at least one mathematics course numbered 3000 or above, or they must obtain permission from the instructor.

Students are also expected to be comfortable with proofs, which means they should have experience in constructing and understanding mathematical proofs. This indicates that the course will likely involve advanced mathematical concepts that require a strong foundation in proof-based mathematics.

In summary, to enroll in this course, students must have completed the prerequisite courses or obtained permission from the instructor, and they must be comfortable with constructing and understanding mathematical proofs.

## MATH4530 Topology HELP（EXAM HELP， ONLINE TUTOR）

Throughout, a topological space $X$ is endowed with a topology $\tau$, even if not explicitly mentioned.

1. A collection $\left{A_\alpha\right}$ of subsets of $X$ satisfies the finite intersection property if $\bigcap_{i=1}^n A_{\alpha_i} \neq \emptyset$ for any finite subcollection.
(a) Prove Cantor’s “finite intersection lemma”: Suppose $\left{K_\alpha\right}$ is a collection of compact sets of a Hausdorff space $X$. If $\bigcap_{i=1}^n K_{\alpha_i} \neq \emptyset$ for any finite subcollection, then $\bigcap_\alpha K_\alpha \neq$ 0.
(b) Prove that $X$ is compact if and only if every collection of closed sets $\left{F_\alpha\right}$ satisfying

(a) To prove Cantor’s “finite intersection lemma”, suppose that ${K_\alpha}$ is a collection of compact sets in a Hausdorff space $X$, and that $\bigcap_{i=1}^n K_{\alpha_i} \neq \emptyset$ for any finite subcollection. We wish to show that $\bigcap_\alpha K_\alpha \neq \emptyset$.

Suppose, for the sake of contradiction, that $\bigcap_\alpha K_\alpha = \emptyset$. Then, for each $\alpha$, there exists an open set $U_\alpha$ such that $K_\alpha \subseteq U_\alpha$ and $\bigcap_\alpha U_\alpha = \emptyset$. Since $X$ is a Hausdorff space, for each pair of distinct points $x,y \in X$, there exist disjoint open sets $U_x$ and $U_y$ containing $x$ and $y$, respectively. For each $\alpha$, let $x_\alpha \in K_\alpha$ be any point, and let $U_{x_\alpha}$ be an open set containing $x_\alpha$ such that $U_{x_\alpha} \subseteq U_\alpha$. Then, ${U_{x_\alpha}}$ is an open cover of $\bigcup_\alpha K_\alpha$.

Since each $K_\alpha$ is compact, there exists a finite subcollection ${U_{x_{\alpha_1}}, \dots, U_{x_{\alpha_n}}}$ that covers $\bigcup_{i=1}^n K_{\alpha_i}$. But then, by construction, $\bigcap_{i=1}^n U_{x_{\alpha_i}} \neq \emptyset$, which contradicts the fact that $\bigcap_\alpha U_\alpha = \emptyset$. Therefore, we must have $\bigcap_\alpha K_\alpha \neq \emptyset$, as desired.

1. On HW 2, you proved that if a function $f: X \rightarrow Y$ between metric spaces is continuous, then its graph
$$\Gamma_f:={(x, f(x)) \mid x \in X}$$
is a closed subset of $X \times Y$. Now, suppose $f: X \rightarrow Y$ is a map between topological spaces, and $Y$ is Hausdorff.
(a) Show that if $f$ is continuous, then the graph $\Gamma_f$ is closed in $X \times Y$.
(b) Show that the conclusion of Part (a) may fail of $Y$ is not Hausdorff.
(c) Show that if $X$ and $Y$ are both compact and Hausdorff, then the converse to Part (a) holds.

(a) Suppose $f$ is continuous and let $(x,y)\in \overline{\Gamma_f}$, i.e., $(x,y)$ is a limit point of $\Gamma_f$. We want to show that $(x,y)\in \Gamma_f$, i.e., $y=f(x)$. Since $(x,y)$ is a limit point of $\Gamma_f$, there exists a sequence $(x_n,y_n)\in \Gamma_f$ such that $(x_n,y_n)\to (x,y)$. Since $Y$ is Hausdorff, we have $y_n\to y$. Since $f$ is continuous, we have $f(x_n)\to f(x)$. But $(x_n,y_n)\in \Gamma_f$, so $y_n=f(x_n)$ for all $n$. Therefore, $y=f(x)$, and $(x,y)\in \Gamma_f$. Hence, $\overline{\Gamma_f}\subseteq \Gamma_f$, so $\Gamma_f$ is closed in $X\times Y$.

(b) Let $X=Y={0,1}$ with the indiscrete topology, i.e., the only open sets are $\emptyset$ and $X$. Let $f:X\to Y$ be the identity map. Then $f$ is continuous, but $\Gamma_f={(0,0),(1,1)}$ is not closed in $X\times Y$.

(c) Suppose $X$ and $Y$ are both compact and Hausdorff, and suppose $\Gamma_f$ is closed in $X\times Y$. We want to show that $f$ is continuous. Let $x_0\in X$ and let $\epsilon>0$. We want to find a neighborhood $U$ of $x_0$ such that $f(U)\subseteq B(f(x_0),\epsilon)$, where $B(f(x_0),\epsilon)$ is the open ball of radius $\epsilon$ centered at $f(x_0)$ in $Y$. Since $Y$ is Hausdorff, for each $x\in X\setminus{x_0}$, there exist disjoint open sets $U_x$ and $V_x$ in $Y$ such that $f(x_0)\in U_x$ and $f(x)\in V_x$. The sets $U_x$ cover $f(x_0)$, so by compactness, there exist finitely many $x_1,\dots,x_n$ such that $U_{x_1},\dots,U_{x_n}$ cover $f(x_0)$. Let $V=V_{x_1}\cap\dots\cap V_{x_n}$, which is an open neighborhood of $x_0$ in $X$. We claim that $f(V)\subseteq B(f(x_0),\epsilon)$. To see this, let $y\in f(V)$, so there exists $x\in V$ such that $f(x)=y$. Then $x\in U_{x_i}$ for some $i$, so $f(x_i)\in U_{x_i}$ and $f(x)=f(x_i)$. Therefore, $y=f(x_i)\in U_{x_i}$, so $U_{x_i}\cap B(f(x_0),\epsilon)\neq\emptyset$. Since $U_{x_i}$ and $V_{x_i}$ are disjoint, we have $f(x_0)\notin V_{x_i}$, so $y=f(x_i)\notin V_{x_i}$.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

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