### Test 3 Study Guide

This test will be cummulative, so review the test 1
study guide and test 2 study guide as
well as both tests.
In addition:
**Additional Definitions**

Cover
Compactness
The statement of Heine-Borel
Be sure to know all the definitions on the study guides from tests 1 and 2
[many of these should hopefully feel very familiar by now]
**Additional Examples**

Examples of finite and infinite topological spaces that are compact and
why.
Examples of topologicial spaces that are not compact and why.
An example of a bounded set in a metric space that is not compact and why.
An example of a closed set in a metric space that is not compact and why.
Compactness in the Cantor set
Compactness in the discrete topology
Compactness in the cofinite topology
Compactness of subsets of R - be able to produce covers that have no
finite subcovers
All the surfaces that locally look like the plane.
Be sure to know all the examples on the study guides from tests 1 and 2
[many of these should hopefully feel very familiar by now]
**Additional Proofs**

Prove that an infinite discrete topology is not compact
Prove that a set with the cofinite topology is compact
Prove that a continuous surjection preserves compactness
Be sure to know all the proofs on the study guides from tests 1 and 2
[there were 3 proofs on each study guide]
**Big Picture**

Why topology was important historically
Why a notion like compactness is important
How did we prove that [0,1] is compact [big picture in one or two
sentences - not the entire proof]