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E85

[0KZ] Prerequisites:[0J1],[026].Conversely, let \(X\) be a set and \({\mathcal B}\) a family of subsets that verify the previous properties (a),(b) seen in [0KX]. Let \(𝜎\) the family of sets that are obtained as a union of elements of \({\mathcal B}\), in symbols 1

\[ 𝜎{\stackrel{.}{=}}\left\{ ⋃_{i∈ I} A_ i : I~ \text{ family of indexes and } ~ A_ i∈ {\mathcal B} βˆ€ i∈ I\right\} ~ ~ ; \]

it is meant that also \(βˆ…βˆˆ 𝜎\). Show that \(𝜎\) is a topology.

Solution 1

[0M0]

  1. As already discussed in [026], you could also use the more compact notation \(𝜎{\stackrel{.}{=}}\left\{ \underline⋃ {\mathcal F} : {\mathcal F} βŠ† {\mathcal B}\right\} \).
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Authors: "Mennucci , Andrea C. G." .
Bibliography
Book index
  • space, topological
  • topological space
  • \( \underline \bigcup \)
  • base, (topology)
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