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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