### 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)=x2 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 R2 and are homeomorphic.
• Two spaces that are subspaces of R2 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