### Test 2 Study Guide

This test will be cummulative, so review the test 1
study guide and
the first test.
**Definitions**

Definitions from the test 1 study guide and
Cartesian Product on Sets and on Topologies
Subspace Topology
Definition of Closed
Definition of Hausdorff
Definition of the Cantor Set
Numerous definitions of continuous
Definition of 1-1
Definition of onto
Definition of homeomorphism
Definition of connected
Definition of a separation
**Examples**

Study guide 1 examples
Open sets and closed sets
in the various topologies from the test 1 study guide, and the Cantor set.
Spaces that are Hausdorff and explanations why
The pictorial argument that f(x)=x^{2} is continuous.
Spaces that are not Hausdorff and explanations why
Spaces that are not metrizable and are not Hausdorff
A space that is not metrizable but is Hausdorff
Spaces that are connected and expanations why
Spaces that are not connected and explanations why
Two spaces that are subspaces of R and are homeomorphic and an explanation
of why.
Two spaces that are subspaces of R and are not homeomorphic and an
explanation of why.
Two spaces that are subspaces of R^{2} and are homeomorphic.
Two spaces that are subspaces of R^{2} and are not homeomorphic.
Whether R_cf with the finite complement topology is homeomorphic to
various spaces
**Proofs**

Proofs from study guide 1
X is discrete iff every function f : X-->R is continuous
If X is metrizable then it is Hausdorff
S^1 and [0,1) are not homeomorphic