Students should develop and exhibit the following abilities:
- Exhibit an ability to understand basic concepts of set theory.
- Exhibit an ability to understand basic concepts of relations.
- Exhibit an ability to understand basic concepts of functions.
- Exhibit an ability to understand basic concepts of sequences, convergence, limits, and continuity for
the real numbers and the absolute value metric.
- Exhibit an ability to understand the concept of distance between sets in a metric space.
- Exhibit an ability to understand the relationship between a subbasis, a basis, and a Topology.
- Exhibit an ability to understand the relationship between a neighborhood system and a Topology.
- Exhibit an ability to understand the concepts of interion and closure of a set.
- Exhibit an ability to understand the relationship between the concepts of open, closed,
closure, and boundary for general topological spaces.
- Exhibit an ability to understand the concept of a subspace and the impact on
derived sets and continuity.
- Exhibit an ability to understand the separation axioms in particular spaces.
- Exhibit an ability to understand the concept of an open cover and compactness.
- Exhibit an ability to understand the concept of connected.
- Demonstrate ability to prove that a relation is an equivalence relation.
- Demonstrate ability to identify if a function is 1-1 or not.
- Demonstrate ability to identify if a function is onto or not.
- Demonstrate ability to show that a set is finite.
- Demonstrate ability to show that a set is countably infinite.
- Demonstrate ability to show that a set is uncountable.
- Demonstrate ability to prove that R is uncountable.
- Demonstrate ability to prove that (0,1) is uncountable.
- Demonstrate ability to prove that a function is a metric for a space.
- Demonstrate ability to show that two metrics have the same collection of open spaces .
- Demonstrate ability to show the convergence of a given sequence in a given metric space.
- Demonstrate ability to discuss the continuity of a given function in a given metric space.
- Demonstrate ability to show if a given collection of sets is a Topology for a given space.
- Demonstrate ability to find the interior of a given set.
- Demonstrate ability to find the closure of a given set.
- Demonstrate ability to find the boundary of a given set.
- Demonstrate ability to prove that a given set is compact in a space.