Last modified on 17 April 2014, at 23:04

Topology/Continuity and Homeomorphisms

Topology
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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 X,Y be topological spaces.

A function f:X\to Y is continuous at x\in X if and only if for all open neighborhoods B of f(x), there is a neighborhood A of x such that A\subseteq f^{-1}(B).
A function f:X\to Y is continuous in a set S if and only if it is continuous at all points in S.

The function f:X\to Y is said to be continuous over X if and only if it is continuous at all points in its domain.

f:X\to Y is continuous if and only if for all open sets B in Y, its inverse f^{-1}(B) is also an open set.
Proof:
(\Rightarrow)
The function f:X\to Y is continuous. Let B be an open set in Y. Because it is continuous, for all x in f^{-1}(B), there is a neighborhood x\in A\subseteq f^{-1}(B), since B is an open neighborhood of f(x). That implies that f^{-1}(B) is open.
(\Leftarrow)
The inverse image of any open set under a function f in Y is also open in X. Let x be any element of X. Then the inverse image of any neighborhood B of f(x), f^{-1}(B), would also be open. Thus, there is an open neighborhood A of x contained in f^{-1}(B). Thus, the function is continuous.


If two functions are continuous, then their composite function is continuous. This is because if f and g have inverses which carry open sets to open sets, then the inverse g^{-1}(f^{-1}(x)) would also carry open sets to open sets.

ExamplesEdit

  • Let X have the discrete topology. Then the map f:X \rightarrow Y is continuous for any topology on Y.
  • Let X have the trivial topology. Then a constant map g:X \rightarrow Y is continuous for any topology on Y.

HomeomorphismEdit

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

DefinitionEdit

Let X,Y be topological spaces
A function f:X\to Yis said to be a homeomorphism if and only if

(i) f is a bijection
(ii) f is continuous over X
(iii)f^{-1} is continuous over Y

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

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

NotesEdit

  1. A map may be bijective and continuous, but not a homeomorphism. Consider the bijective map f:[0,1) \rightarrow S^1, where f(x)=e^{2\pi ix} 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 S^1, like the set \left[ 0,\frac{1}{2}\right).
  2. Homeomorphism is an equivalence relation

ExercisesEdit

  1. Prove that the open interval (a,b) is homeomorphic to \mathbb{R}.
  2. Establish the fact that a Homeomorphism is an equivalence relation over topological spaces.
  3. (i)Construct a bijection f:[0,1]\to [0,1]^2
    (ii)Determine whether this f is a homeomorphism.


Topology
 ← Quotient Spaces Continuity and Homeomorphisms Separation Axioms →