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/

Course home page

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 $\R\times \R$.