Topological space

Definition

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Consider   to be a non-empty set, and also let   be a subset of the power set of  , such that an action   fullfils the following conditions,

  •  ,
  • if   then also the finite intersetion of these sets are element of the topology, i.e.
 .
  • let   be an index set and for all   the subset   is element of the topology ( ) then also the union of these sets   is an element of the topology <\math>, i.e.
 .

The pair   is called topological space. Set sets in   are called the open sets in  .

Learning Task

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  • Let   and  . Add a minimal number of sets, so   and create  , so that   is a topological space.