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Non-Hausdorff Topology and Domain Theory by Jean Goubault-Larrecq

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4

Topology

4.1 Topology, topological spaces

Abstracting away from metrics, a topological space is a set, with a collection of so-called open subsets U, satisfying the following properties. We have already seen them, for sequentially open subsets of a metric space, in Proposition 3.2.7.

Definition 4.1.1 (Topology) Let X be a set. A topology on X is a collection of subsets of X, called the opens of the topology, such that:

  • every union of opens is open (including the empty union, Ø);
  • every finite intersection of opens is open (including the empty intersection, taken as x itself).

A topological space is a pair (X, ), where is a topology on X.

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