## Math 321 General Topology, 2018 Fall

### Instructor: Professor Jae Choon Cha

Office hour: by an appointment
Instructor’s web page: http://gt.postech.ac.kr/~jccha/

http://gt.postech.ac.kr/~jccha/general-topology-2018-fall/

### Classroom and hour

Monday and Wednesday 11:00–12:15pm, Math Building 402

### Syllabus

A brief version is available from POVIS. A more detailed pdf version is available here.

### Homework Problems

Homework problems will be posted on this web page every week, and will be collected in class on the due date. Turn it in BEFORE the lecture starts.  $\def\C{\mathbb{C}}\def\R{\mathbb{R}}\def\Q{\mathbb{Q}}\def\Z{\mathbb{Z}}\def\T{\mathcal{T}}\def\B{\mathcal{B}}\def\inte{\mathop{\text{int}}}$

#### Problem Set #1

Due 09/27 23:59pm. Hand in your solution to the Math 321 homework box on the 1st floor in the math building.

Read Munkres p.75-88 (you may skip Lemma 13.4), p.102-108 (not including Theorem 18.1 and 18.3 for now).

1. Suppose $X$ is a set.
1. Let $\T$ be the collection of subsets $U$ of $X$ such that $X-U$ is either finite or equal to $X$. Show that $\T$ is a topology on $X$.  ($\T$ is called the finite complement topology.)
2. Let $\T$ be the collection of subsets $U$ of $X$ such that $X-U$ is either countable or equal to $X$. Show that $\T$ is a topology on $X$.
3. Let $\T$ be the collection of subsets $U$ of $X$ such that $X-U$ is either infinite, empty or equal to $X$. Is $\T$ a topology? Prove or disprove by a counterexample.
2. Let $\B=\{(a,b) \subset \R \mid a,b\in \Q,\; a< b\}$ and $\B’=\{(a/2^n,b/2^n) \subset \R \mid a,b,n\in \Z,\; a< b\}$.
1. Show that both $\B$ and $\B’$ are bases on $\R$.
2. Show that $\B$ and $\B’$ generate the same topology on $\R$.
3. Suppose $\{\T_\alpha\}$ is a collection of topologies $\T_\alpha$ on a fixed set $X$.
1. Show that $\bigcap_\alpha \T_\alpha$ is a topology on $X$.
2. Is $\bigcup_\alpha \T_\alpha$ a topology? Prove or disprove by a counterexample.
4. For two topologies $\T$ and $\T’$ on a set $X$, if $\T \subset \T’$, then we say that $\T$ is smaller than $\T’$ (and $\T’$ is larger than $\T$; often we also say $\T$ is coarser than $\T’$ and $\T’$ is finer than $\T$).
1. What is the largest topology among all the topologies on a fixed set $X$?
2. Suppose $\{\T_\alpha\}$ is a collection of topologies on a set $X$. Show that there is a unique largest topology contained in every $\T_\alpha$.
3. Suppose $\B$ is a base for a topology on $X$. Show that the topology $\T$ generated by $\B$ is the smallest topology among the topologies on $X$ containing $\B$.  (In order words, $\T$ contains $\B$, and if $\T’$ is a topology containing $\B$, then $\T$ is smaller than $\T’$.)
5. Suppose $X$ and $Y$ are spaces and $y_0\in Y$. Let $c\colon X\to Y$ be the constant function defined by $c(x)=y_0$. Show that $c$ is continuous.
6. Give a rigorous proof that any two (nonempty) open intervals $(a,b)$ and $(c,d)$ in $\R$ are homeomorphic.  Here $(a,b)$ and $(c,d)$ are equipped with the order topology.
7. The dictionary order on $\R\times\R$ is defined as follows: $(x,y)<(x’,y’)$ if and only if either $x<x’$, or $x=x’$ and $y<y’$.  Let $\R_d$ be the set $\R$ equipped with the discrete topology. Show that the product topology on $\R_d\times \R$ is equal to the dictionary order topology on $\R\times \R$.