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Math and science::Topology

Topology and topological space. Definition

A topology on a set \( X \) is a collection \( \mathcal{T} \) of subsets of \( X \) with the following properties:

T1
Whenever \( (U_i)_{i \in I} \) is a family (finite or not) of subsets of \( X \) such that \( U_i \in \mathcal{T} \) for all \( i \in I \), then  [...].
T2
Whenever \( U_1, U_2 \in \mathcal{T} \), then [...].
T3
[\( \text{something} \in \mathcal{T} \)] and [\( \text{something} \in \mathcal{T} \)].

A topological space \( (X, \mathcal{T}) \) is a pair consisting of a set \( X \) together with a topology \( \mathcal{T} \) on \( X \).

T3 can be derived from T1 and T2, and is not strictly necessary as an axiom; however, most treatments of the subject introduce the definition as this triplet of statements.

In relaxed words

T1 can be phrased as 'an arbitrary union of open subsets is open'. By induction T2 can be phrased as 'a finite intersection of open subsets is open'.

Combining these two then, a topology on \( X \) can be defined as 'a collection of [something] that is [something something] arbitrary unions and finite intersections'.