# Topology Qualifying Exam Workshop

May 2020

Worksheet 2

Theme: Topological spaces and continuous functions: Topology, open and closed sets, basis, sub- basis; continuous function, homeomorphism; closure, limit points; subspace topology, product topology, and quotient/identification topology.

Also some compactness because there weren’t a ton of problems on just those topics.

Part 1 - 5-10 minutes

Warm-up:

1. What is a topology on a set?

2. Describe how to make a topology from a basis, and what (if any) requirements there are on the basis.

3. Describe how to make a topology from a subbasis, and what (if any) requirements there are on the subbasis.

4. Give 4-6 different topologies on R.

## Part 2 - 1 hour and 40 minutes

Try these problems:

1. (June 2010) Let X be a topological space and let $f , g : X \to$ R be continuous functions.

(a) Show that the set $L = \{ p \in X : f ( p ) \leq g ( p ) \}$ is a closed subset of X.

(b) Show that the function $h : X \to \mathbb { R }$ given by $h ( p ) = \operatorname* { m i n } \{ f ( p ) , g ( p ) \}$ is continuous.

2. (June 2011) Prove that if $f : X \to Y$ is a continuous map between topological spaces and C is a compact subset of X, then f (C) is a compact subset of Y .

3. (June 2012) Suppose X and Y are topological spaces and $f : X \to Y$ is a continuous function.

(a) Prove there exists a topological space Z, a quotient map $q : X \to Z$ and a one-to-one continuous map $h : Z \to Y$ such that $f = h \circ q .$

(b) (This part is a continuous of part (a).) If Z1 is a topological space, $q _ { 1 } : X \to Z _ { 1 }$ is a quotient map, $h _ { 1 } : Z _ { 1 } \to Y$ is one-to-one and continuous, and $f = h _ { 1 } \circ q _ { 1 }$ , prove that Z and $Z _ { 1 }$ are homeomorphic.

4. (May 2015) Let $f : ( X , { \mathcal { T } } ) \to ( Y , { \mathcal { T } } ^ { \prime } )$ be a continuous map.
   A section of f is a continuous map $\sigma : ( Y , { \mathcal { T } } ^ { \prime } ) \to ( X , { \mathcal { T } } )$ such that $f \circ \sigma = \operatorname { i d } _ { Y }$ . Show that if f has a section, then f is a quotient map.

5. (May 2020) Let $B = \{ [ a , b ) : a , b \in \mathbb { R } \}$ ; let $\mathcal { R } _ { \ell }$ denote R equipped with the topology generated by the basis B (you do not need to prove B is a basis).
   Decide (with proof) if the following sets are compact in $\mathbb { R } _ { \ell } { : }$

(a) A = {1/n : n ∈ N} ∪ {0}.

(b) $B = \{ - 1 / n : n \in \mathbb { N } \} \cup \{ 0 \}$

## Part 3 - 5-10 minutes

Wrap-up discussion with everyone:

1. Which of these problems did you find easy?
   Hard?

2. Which topics in the theme do you need to review more?