--POINT-SET TOPOLOGY IN EUCLIDEAN SPACE

You should know all definitions and theorems from the course and be able to use them,  
in particular: 

 *be able to identify (with proper explanations) 
  interior, exterior, and frontier points for a given subset A in R^n
  
 *determine (with proofs, arguing from definitions) whether a given set in R^n
  is open or closed (or neither open nor closed). Your arguments should discuss 
  neighborhoods of points, not generic intuitive ideas about whether the set 
  includes boundary etc.
 
 *find (with proofs) the interior, the closure, and the frontier of a given set. 
  Be able to prove (a) the interior of any set is open, (b) the closure of any set 
  is closed, (c) the frontier of any set is closed
  
 *know and be able to prove (in R^n) fundamental properties of open sets: arbitary 
  unions and finite intersections of open sets are open.     
 
 *a set is closed iff its complement is open (know proof)

 * relatively open/closed subsets in a given set A (see below!)  
 
--ABSTRACT POINT-SET TOPOLOGY

 * know definition of topology on an abstract set (as a collection of designated "open" subsets
  such that the empty set and the whole space are open, and arbitary 
  unions and finite intersections of open sets are open; be able to check whether a particular 
  collection of subsets is a topology, know basic examples (topology on R^n, discrete, indiscrete 
  topology)       

 * closed sets are defined as complements of open sets; be able to prove properties (finite unions 
   and arbitrary intersections of closed sets are closed) 

 * definition of a basis (as a collection of neighborhoods satisfying properties in Definition (3.3)
   from the book); be able to check whether a given collection is a basis; know a basis 
   for the usual Euclidean topology in R^n 
 
 * generating a topology from a given basis, either from Definition (3.4) (via "interior" points) 
   or Theorem (3.6) (open sets are unions of subsets from the basis); be able to figure out 
   this topology in simple examples (e.g. does a given basis give a standard topology on R? is 
   a particular given set open in this topology?)

 * equivalent bases (be able to show when two different bases give the same topology)

 * subspace topology on a subset A of a space X, relatively open/closed subsets in A.
  define relatively open sets from the basis of relative neighborhoods in A or as intersections with A 
  of sets open in X; be able to identify whether a given subset B in A is relatively open or 
  closed in A. Important special case: A is a subset of R^n with usual topology.   

 * Product topology (be able to construct a basis, identify open/closed sets)

--CONTINUOUS FUNCTIONS

 * Definition via preimages of open sets; be able to check (with proof), 
   in simple examples, if a given  function between two topological spaces is continuous 
  
 * Useful fact: a function is continuous iff preimages of all sets in a basis for given topology 
   are open. Know proof, be able to use it. (Notice that for R^n, it means you only need to check 
  inverse images of neighborhoods D^n(x,r) to establish continuity)

 * epsilon-delta definition for continuity of functions on R^n. Equivalence of the epsilon-delta
   and open sets approaches. Upshot: familiar continuous functions from analysis are continuous 
   in the sense of topology

 * Properties: composition of continuous functions is continuous. 
   Restrictions of continuous functions: if f:R^n --> R^n is continuous, A a subset of R^n, 
   then the restricted function f: A --> f(A) is also continuous.

 * Homeomorphisms: intuitive idea as topological equivalence; precise definition. Simple 
   examples of homeomorphic spaces: intervals of different length, open interval and line, etc.
   Be able to check whether two given simple (for example, finite) topological spaces 
   are homeomorphic. (For now, we have no special tools, so this is only possible when you
   can compare open sets directly)