Topological Space

A topological space is a set for which closeness can be defined.

Description

A topological space is the double (X, τ) composed of:

• a given space X.

• the topology of X, notated as τ.

The topology of X is the non-empty collection of open subsets of X satisfying these conditions:

• Both the empty set and the full set are in the topology; Ø, X ∈ τ

  • In other words, both the empty set and the full set are definitionally open.

• The topology is closed upon countable unions; ⋃ X_i ∈ τ where members of τ be indexed as X_i, i ∈ N

  • In other words, any countable union of the collection is open.

• The topology is closed upon finite intersections; ⋂ X_i ∈ τ for i from 1 to n

  • In other words, any finite intersection of the collection is open.

Alternate Formulation

A topological space is a set X for which a neighborhood topology function is defined. Generally speaking, this is a function 𝒩(x) that returns a non-empty subset of X representing the neighborhood of x. It must satisfy these conditions:

• a member is always in its own neighborhoods; if N ∈ 𝒩(x) then x ∈ N

• any superset of a neighborhood of x is also a neighborhood of x

• any intersection of two neighborhoods of x is also a neighborhood of x

• for all neighborhoods of x as N, there is another neighborhood of x as M that is a subset of N; furthermore N is a neighborhood for every member of M

Comparison to Metric Spaces

Metric spaces are a similar concept. Their main characteristic is the presence of a distance function, which in some ways is a constraint to application, but it does provide a succinct way to define neighborhoods.

CategoryRicottone