Maths - Topological Space

Topological Space

We want to study the properties of manifolds that are preserved by continuous maps, to do this we do not want to use the concept of distance (metric space) and yet we still need the concept of continuous functions to make sense.

To do this we use the concept of 'open space' and 'open set'. The notion of an open set provides a way to speak of distance in a topological space, without explicitly defining a metric on the space. Although one cannot obtain concrete values for the distance between two points in a topological space, one may still be able to speak of "nearness" in the space, thus allowing concepts such as continuity to translate into the theory of open sets.

Hausdorff Space

A topological space is said to be a Hausdorff space if given any pair of distinct points p₁, p₂ H, there exists neighborhoods U₁ of p₁ and U₂ of p₂ with U₁ U₂ = Ø.

In other words: open sets separate points.

Topology on a Set

A topology on a set X is a collection Τ of subsets of X, called open sets satisfying the following properties:

• X and Ø are elements of Τ.
• Τ is closed under finite intersections.
• Τ is closed under arbitrary unions.

Examples:

	This is not a topological space because 'a' and 'b' are subsets but not the union of 'a' and 'b'
	This is now a topological space because we have added the union of 'a' and 'b'
	This is not a topological space because 'ab' and 'bc' are subsets but not their intersection.
	This is now a topological space because we have added 'b'

Note: Assume Ø is included in the above examples.

Open Set Criterion for Continuity

A map ƒ: M₁ → M₂ is continuous if and only if the inverse image of every open set is open.

Further Information

Manifolds