Math 2.0

definition

Topological space

Definition

Let XX be a set. A topology on XX is a set τ\tau of subsets of XX with these three properties:

  1. The empty set ∅\varnothing and the set XX are in τ\tau.
  2. For each family (Ui)i∈I(U_i)_{i \in I} of members of τ\tau, the union ⋃i∈IUi\bigcup_{i \in I} U_i is in τ\tau.
  3. For all U,V∈τU, V \in \tau, the intersection U∩VU \cap V is in τ\tau.

A topological space is a pair (X,τ)(X, \tau) where XX is a set and τ\tau is a topology on XX.

Remarks

The members of τ\tau are the open sets of the space. Their complements are the closed sets. By induction, property 3 gives that each finite intersection of open sets is open. Infinite intersections need not be open. Each metric gives a topology, the metric topology. Other basic examples are the discrete topology and the indiscrete topology. One can also define a topology by its closed sets, or by the neighbourhoods of its points.

Explanations

L1 human

A topology tells you which subsets of XX count as open, without any distances. The rules copy the properties of open sets in metric spaces. Any union of open sets is open, and a finite intersection of open sets is open.

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Sources

  1. Web https://en.wikipedia.org/wiki/Topological_space, Definition via open sets

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