Topology/Continuity and Homeomorphisms

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Contents

ContinuityEdit

Continuity is the central concept of topology. Essentially, topological spaces have the minimum necessary structure to allow a definition of continuity. Continuity in almost any other context can be reduced to this definition by an appropriate choice of topology.

DefinitionEdit

Let   be topological spaces.

A function   is continuous at   if and only if for all open neighborhoods   of  , there is a neighborhood   of   such that  .
A function   is continuous in a set   if and only if it is continuous at all points in  .

The function   is said to be continuous over   if and only if it is continuous at all points in its domain.

  is continuous if and only if for all open sets   in  , its inverse   is also an open set.
Proof:
( )
The function   is continuous. Let   be an open set in  . Because it is continuous, for all   in  , there is a neighborhood  , since B is an open neighborhood of f(x). That implies that   is open.
( )
The inverse image of any open set under a function   in   is also open in  . Let   be any element of  . Then the inverse image of any neighborhood   of  ,  , would also be open. Thus, there is an open neighborhood   of   contained in  . Thus, the function is continuous.


If two functions are continuous, then their composite function is continuous. This is because if   and   have inverses which carry open sets to open sets, then the inverse   would also carry open sets to open sets.

ExamplesEdit

  • Let   have the discrete topology. Then the map   is continuous for any topology on  .
  • Let   have the trivial topology. Then a constant map   is continuous for any topology on  .

HomeomorphismEdit

When a homeomorphism exists between two topological spaces, then they are "essentially the same", topologically speaking.

DefinitionEdit

Let   be topological spaces
A function  is said to be a homeomorphism if and only if

(i)   is a bijection
(ii)   is continuous over  
(iii)  is continuous over  

If a homeomorphism exists between two spaces, the spaces are said to be homeomorphic

If a property of a space   applies to all homeomorphic spaces to  , it is called a topological property.

NotesEdit

  1. A map may be bijective and continuous, but not a homeomorphism. Consider the bijective map  , where   mapping the points in the domain onto the unit circle in the plane. This is not a homeomorphism, because there exist open sets in the domain that are not open in  , like the set  .
  2. Homeomorphism is an equivalence relation

ExercisesEdit

  1. Prove that the open interval   is homeomorphic to  .
  2. Establish the fact that a Homeomorphism is an equivalence relation over topological spaces.
  3. (i)Construct a bijection  
    (ii)Determine whether this   is a homeomorphism.


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