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; ⋃ Xi ∈ τ where members of τ be indexed as Xi, i ∈ N
- In other words, any countable union of the collection is open.
The topology is closed upon finite intersections; ⋂ Xi ∈ 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.