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