Topology/Continuity and Homeomorphisms

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Contents 1 Continuity 1.1 Definition 1.2 Examples 2 Homeomorphism 2.1 Definition 2.2 Notes 3 Exercises

[edit] Continuity

Continuity is the central concept of topology. Essentially, topological spaces have the minimum necesary 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.

[edit] Definition

Let X,Y be topological spaces.

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

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

is continuous if and only if for all open sets B in Y, its inverse f ^{- 1}(B) is also an open set.
Proof:
()
The function is continuous. Let B be a open set in Y. Because it is continuous, for all x in f ^{- 1}(B), there is a neighborhood , since B is an open neighborhood of f(x). That implies that f ^{- 1}(B) is open.
()
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.

[edit] Examples

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

[edit] Homeomorphism

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

[edit] Definition

Let X,Y be topological spaces
A function is said to be a Homeomorphism if and only if

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

[edit] Notes

1. A map may be bijective and continuous, but not a homeomorphism. Consider the bijective map , where f(x) = e^2πix 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 S¹, like the set .
   
2. Homeomorphism is an equivalence relation

[edit] Exercises

1. Prove that the open interval (a,b) 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 f is a homeomorphism.