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Illustration Elements of a topology

Elements of a topology
Total set and empty set as elements of a topology
Intersections as elements of a topology
Union sets as elements of a topology
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The elements of a topology \(\class{blue}{\mathcal{T}}\) of a topological space \( (\mathbb{X}, \class{blue}{\mathcal{T}}) \) must satisfy three properties:

  1. The empty set \( \emptyset \) and the whole set \(X\) are both inside \(\mathcal{T}\).
  2. The union of any (even infinitely many) elements \(\mathbb{T}_i\) of \(\class{blue}{\mathcal{T}}\) is also an element of \(\class{blue}{\mathcal{T}}\).
  3. The intersection of finitely many elements \(\mathbb{T}_i\) of \(\class{blue}{\mathcal{T}}\) is also an element of \(\class{blue}{\mathcal{T}}\).