So in the first lesson we learned what a topology is, what open sets, closed sets, and bases are. You should be comfortable with these concepts before beginning this second lesson.

Continuous Function

edit

We will define the notion of a continuous function below. (Note that in topological texts and papers it is common to use the word 'map' and even the word 'function' to mean a continuous function. To avoid ambiguity, in this course we will reserve the word 'function' to mean any function, but will use 'map' to mean a continuous function.)

Definition

edit

Let   and   be topological spaces. A function   is called continuous if for every   we have   That is, f is continuous iff the f-preimage of every open set (in  ) is open (in  ).

Examples

edit

It is immediate from the definition that the following two types of functions are always continuous. The proof of these two claims is left as an exercise.

  1. If   is a discrete space and   is any space, then any function   is continuous.
  2. If   is any space and   has the indiscrete topology, then any function   is continuous.

Continuous at a point

edit

It is also possible to talk about a function being continuous at a point of its domain. So, given a map   and a point   we say that   is continuous at   if given any neighborhood   of   there is a neighborhood   of   such that  

Exercises

edit

Let   be a function. Show that the following are equivalent.

  1.   is continuous.
  2.   is continuous at   for all  
  3. For any closed set   we have   is closed in  
  4. If   is a basis for   then for any set   we have   is open in  

Open maps

edit

The definition of a continuous map may seem awkward. Since it is a morphism in the category of topological spaces, one would expect it to preserve some property about open sets, but what one might first think is that open sets are preserved under the map of the function. But this gives a different concept.

Definition (open map)

edit

Let   be a continuous function. Then we say that   is an open map if for any open set   we have   is open in  

Merely for the purposes of the discussion here, define an open function to be a function (not necessarily continuous)   such that   is open in   whenever   is open in  

Exercises

edit
  1. Construct finite-point spaces   and   and a map   that is continuous but not open.
  2. Construct another function   that is not continuous but is an 'open function'.
  3. Show that the identity map   is always continuous and open.
  4. Suppose that   and   are two distinct topologies on the set   Suppose that the identity map   is continuous. Show that   In this case, we say that   is finer than   or that   is coarser than  


Homeomorphism

edit

The "isomorphism" or "equivalence" of topological spaces is called "homeomorphism." This is analogous to bijection in the case of sets and group isomorphism in the case of groups. Topologically speaking, two spaces are indistinguishable if they are homeomorphic.

Definition

edit

Let   Then we say that   is a homeomorphism if it is bijective and both   and   are continuous. In this case we say that   and   are homeomorphic and sometimes write   or  

Examples

edit
  1. If   is any topological space then the identity map   is a homeomorphism.
  2. If   is injective and   and   are both continuous then   is called an embedding. In this case the map   given by   is a homeomorphism. That is, an embedding is a homeomorphism with its image in the target space.
  3. The map   given by   is a homeomorphism.

Exercises

edit
  1. Show that the maps given in the examples are indeed homeomorphisms.
  2. Show that if   is a homeomorphism then   is also a homeomorphism.
  3. Show that if   is homeomorphic to   and   is homeomorphic to   then   is homeomorphic to   Note that these first 3 exercises show that homeomorphism is an equivalence relation.

Next Lesson

Notes

edit