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.
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.)
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 ).
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.
- If is a discrete space and is any space, then any function is continuous.
- If is any space and has the indiscrete topology, then any function is continuous.
Continuous at a point
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
Let be a function. Show that the following are equivalent.
- is continuous.
- is continuous at for all
- For any closed set we have is closed in
- If is a basis for then for any set we have is open in
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)
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
- Construct finite-point spaces and and a map that is continuous but not open.
- Construct another function that is not continuous but is an 'open function'.
- Show that the identity map is always continuous and open.
- 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
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.
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
- If is any topological space then the identity map is a homeomorphism.
- 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.
- The map given by is a homeomorphism.
- Show that the maps given in the examples are indeed homeomorphisms.
- Show that if is a homeomorphism then is also a homeomorphism.
- 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.